Report NASA Grant NSG 1581, U. Missouri-Rolla, Dept. of Mech. and Aero. .... 64] William A. Page, Dale L. Compton, William J. Borucki, Donald L. Ci one, and.
Nonequilibrium Radiative Heating Prediction Method for Aeroassist Flow elds with Coupling to Flow eld Solvers by
Lin C. Hartung A Thesis Submitted to the Graduate Faculty of North Carolina State University in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy Major Subject: Aerospace Engineering
Department of Mechanical and Aerospace Engineering Raleigh
1991 Copyright c 1991 by Lin C. Hartung.
Nonequilibrium Radiative Heating Prediction Method for Aeroassist Flow elds with Coupling to Flow eld Solvers by
Lin C. Hartung A Thesis Submitted to the Graduate Faculty of North Carolina State University in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy Major Subject: Aerospace Engineering
Department of Mechanical and Aerospace Engineering Raleigh 1991
Approved by: ______________________________
______________________________
______________________________
______________________________
______________________________ Chairman of Advisory Committee
Abstract Hartung, Lin C. Nonequilibrium Radiative Heating Prediction Method for Aeroassist Flow elds with Coupling to Flow eld Solvers (Under the direction of Dr. F. R. DeJarnette)
A method for predicting radiation absorption and emission coecients in thermochemical nonequilibrium ows is developed. The method is called LORAN: the Langley Optimized RAdiative Nonequilibrium code. It applies the smeared band approximation for molecular radiation to produce moderately detailed results and is intended to ll the gap between detailed but costly prediction methods, and very fast but highly approximate methods. The optimization of the method to provide ecient solutions allowing coupling to ow eld solvers is discussed. Representative results are obtained and compared to previous nonequilibrium radiation methods, as well as to ground- and ight-measured data. Reasonable agreement is found in all cases. A multi-dimensional radiative transport method is also developed for axisymmetric ows. Its predictions for wall radiative ux are 20 to 25 percent lower than those of the tangent slab transport method, as expected, though additional investigation of the symmetry and out ow boundary conditions is indicated. The method has been applied to the peak heating condition of the Aeroassist Flight Experiment (AFE) trajectory, with results comparable to predictions from other methods. The LORAN method has also been applied in conjunction with the computational uid dynamics (CFD) code LAURA to study the sensitivity of the radiative heating prediction to various models used in nonequilibrium CFD. This study suggests that radiation measurements can provide diagnostic information about the detailed processes occurring in a nonequilibrium ow eld because radiation phenomena are very sensitive to these processes.
Acknowledgments
ii
The assistance and suggestions of Dr. Hassan A. Hassan in developing the radiative transport solution method were most valuable, while the comments of Dr. Fred DeJarnette helped me to maintain my perspective on the problem at hand. Thanks go to Drs. Peter Gno o and Robert Mitcheltree, of NASA Langley Research Center, for their patient answers to my many questions regarding numerical methods. In addition, Dr. Mitcheltree's assistance in coupling the LAURA and LORAN solution methods was invaluable. The support of my management at NASA Langley during this period of research was crucial to my success. Flow eld solutions for this study were provided by Robert Mitcheltree, Peter Gno o and Bob Greendyke. Special thanks go to Tad Guy for applying his considerable computer wizardry (and patience) to the sundry bugs and problems I encountered. I would also like to thank Dr. Ann Carlson for her support and her example. I greatly appreciated the encouragement of soon-to-be Drs. Neil Cheatwood, Chris Riley and Tom Kashangaki. In addition, I want to acknowledge the many friends, acquaintances and family members who were interested enough to ask again and again over a period of years about the progress of my e orts. The support and encouragement of my entire family, as well as some friendly competition with my brother Walter, was very valuable in helping me complete this work as quickly as possible.
Summary
iii
When a spacecraft enters the atmosphere of a planet, it pushes some of the gas particles it encounters with it. As the atmosphere becomes thicker, the gas particles in this \shock layer" begin to collide with each other. This converts some of the energy of motion that these particles gained from the spacecraft into internal motions of the electrons and atoms inside the gas particles. When the gas density is high enough, this process reaches an equilibrium state and can be simply described. When the gas density is not so high, the transfer of energy proceeds slowly and remains out of equilibrium. The internal changes in electron and atom energy can be accompanied by a photon of radiative energy being emitted or absorbed by a gas particle. Predicting the number and energy of these photons in a nonequilibrium gas is the problem to be solved here. These photons travel through the gas at the speed of light and may end up depositing their energy on the spacecraft surface, thus adding to the heating which exists because of gas friction. If enough energy is deposited the surface may begin to melt or burn. Heating predictions allow the spacecraft heatshield to be designed accordingly.
iv
Table of Contents List of Figures
vii
List of Tables
x
Nomenclature
xii
1 Introduction
1
2 Historical Review
5
2.1 Topical and Literature Survey : : : : : : : : : : : : : : : : : : : : : :
5
2.1.1 Nonequilibrium Absorption and Emission : : : : : : : : : : : :
5
2.1.2 Nonequilibrium Excitation Processes : : : : : : : : : : : : : :
8
2.1.3 Radiative Transport : : : : : : : : : : : : : : : : : : : : : : :
9
2.2 Computer Code Survey : : : : : : : : : : : : : : : : : : : : : : : : : :
12
2.2.1 Equilibrium Radiation Codes : : : : : : : : : : : : : : : : : :
12
2.2.2 Nonequilibrium Radiation Codes : : : : : : : : : : : : : : : :
14
2.3 Survey of Experimental Results : : : : : : : : : : : : : : : : : : : : :
15
2.3.1 Ground-Based Data : : : : : : : : : : : : : : : : : : : : : : : :
15
2.3.2 Flight Data : : : : : : : : : : : : : : : : : : : : : : : : : : : :
16
3 Radiation Theory
17
v 3.1 Absorption : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
17
3.1.1 Atomic Mechanisms : : : : : : : : : : : : : : : : : : : : : : :
17
3.1.2 Molecular Mechanisms : : : : : : : : : : : : : : : : : : : : : :
22
3.2 Emission : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
34
3.2.1 Atomic Mechanisms : : : : : : : : : : : : : : : : : : : : : : :
35
3.2.2 Molecular Mechanisms : : : : : : : : : : : : : : : : : : : : : :
38
3.3 Induced Emission : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
40
4 Transport Theory
42
4.1 Flow eld Coupling : : : : : : : : : : : : : : : : : : : : : : : : : : : :
43
4.2 Plane-Parallel Medium : : : : : : : : : : : : : : : : : : : : : : : : : :
43
4.2.1 Transparent Gas : : : : : : : : : : : : : : : : : : : : : : : : :
44
4.2.2 Absorbing Gas : : : : : : : : : : : : : : : : : : : : : : : : : :
45
4.3 Multi-Dimensional Medium : : : : : : : : : : : : : : : : : : : : : : :
47
5 Implementation
52
5.1 Coding : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
52
5.1.1 Initial Setup : : : : : : : : : : : : : : : : : : : : : : : : : : : :
52
5.1.2 Nonequilibrium Excitation : : : : : : : : : : : : : : : : : : : :
58
5.1.3 Radiation Properties : : : : : : : : : : : : : : : : : : : : : : :
59
5.1.4 Radiative Transport : : : : : : : : : : : : : : : : : : : : : : :
63
5.2 Computational Optimization : : : : : : : : : : : : : : : : : : : : : : :
66
5.2.1 Radiation Calculation : : : : : : : : : : : : : : : : : : : : : :
67
vi 5.2.2 Excitation Calculation : : : : : : : : : : : : : : : : : : : : : :
68
5.2.3 Radiation Subgrid : : : : : : : : : : : : : : : : : : : : : : : :
70
5.3 Flow eld Coupling : : : : : : : : : : : : : : : : : : : : : : : : : : : :
73
6 Results and Discussion
76
6.1 Comparison to Experiment : : : : : : : : : : : : : : : : : : : : : : : :
76
6.1.1 Ground-Based Data: AVCO Shock Tube : : : : : : : : : : : :
76
6.1.2 Flight Data: Project FIRE : : : : : : : : : : : : : : : : : : : :
77
6.2 Nonequilibrium Test Cases : : : : : : : : : : : : : : : : : : : : : : : :
85
6.2.1 Mars Return : : : : : : : : : : : : : : : : : : : : : : : : : : : :
85
6.2.2 Aeroassist Flight Experiment : : : : : : : : : : : : : : : : : :
90
7 Flow eld Model Studies
95
7.1 Baseline Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
96
7.2 Energy Relaxation : : : : : : : : : : : : : : : : : : : : : : : : : : : :
97
7.2.1 Dissociation Temperature : : : : : : : : : : : : : : : : : : : :
98
7.2.2 Vibrational-Translational Energy Exchange : : : : : : : : : : :
99
7.2.3 Energy Exchange in Dissociation : : : : : : : : : : : : : : : : 104 7.3 Spectral Radiation Comparison : : : : : : : : : : : : : : : : : : : : : 108 7.4 Nonequilibrium Chemistry : : : : : : : : : : : : : : : : : : : : : : : : 112 7.5 Nonequilibrium Temperature : : : : : : : : : : : : : : : : : : : : : : : 115
8 Conclusions
118
vii 8.1 Accomplishments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 118 8.2 Future Work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 119
References
121
A Gaussian Units
135
B Finite Volume Formulation of Radiative Transport
136
B.1 Finite Volume Development : : : : : : : : : : : : : : : : : : : : : : : 136 B.2 Boundary Conditions : : : : : : : : : : : : : : : : : : : : : : : : : : : 147 B.2.1 Axis of Symmetry Boundary : : : : : : : : : : : : : : : : : : : 148 B.2.2 Freestream Boundary : : : : : : : : : : : : : : : : : : : : : : : 151 B.2.3 Out ow Boundary : : : : : : : : : : : : : : : : : : : : : : : : 153 B.2.4 Wall Boundary : : : : : : : : : : : : : : : : : : : : : : : : : : 154 B.2.5 Radiative Heat Flux to Wall : : : : : : : : : : : : : : : : : : : 155 B.3 Numerics and Convergence : : : : : : : : : : : : : : : : : : : : : : : : 155 B.3.1 Convergence Criterion : : : : : : : : : : : : : : : : : : : : : : 155 B.3.2 Selection of Relaxation Parameter : : : : : : : : : : : : : : : : 157
viii
List of Figures 3.1 Distribution of Spectral Points for an Atomic Line : : : : : : : : : : :
18
4.1 Shock Layer Geometry for Plane-Parallel Approximation : : : : : : :
44
5.1 Example of Spectrum Optimization for Bound-Free Continuum : : : :
55
5.2 Example of Spectrum Optimization for a Vibrational Band : : : : : :
57
5.3 Occurrence of Negative 0 in a Test Flow eld : : : : : : : : : : : : :
66
5.4 Nonequilibrium Population of Excited States of O for Negative 0 : :
67
5.5 Example of Radiation Subgrid Selection : : : : : : : : : : : : : : : :
71
5.6 Patchwork Grid Resulting from Application of Subgrid Algorithm Along Each Normal Line : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
73
6.1 Calculated Spectrum for the Peak Radiation Point of the AVCO Experiment : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
78
6.2 Measured Spectrum and NEQAIR Prediction for the Peak Radiation Point of the AVCO Experiment : : : : : : : : : : : : : : : : : : : : :
78
6.3 Predicted FIRE II Stagnation Line Temperature Pro les - 1631 sec :
81
6.4 Predicted FIRE II Stagnation Line Temperature Pro les - 1634 sec :
81
6.5 Predicted FIRE II Stagnation Line Temperature Pro les - 1637.5 sec
82
6.6 Emission Pro les for FIRE II at 1634 sec with Comparison to NEQAIR 83 6.7 Emission Pro les for FIRE II at 1637.5 sec with Comparison to NEQAIR 84
ix 6.8 Spectral Variation of Stagnation Point Radiative Heat Transfer for FIRE II : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
86
6.9 Spectral Variation of Stagnation Point Nitrogen Radiation for FIRE II 86 6.10 Wall Radiative Flux for Mars Return Case : : : : : : : : : : : : : : :
88
6.11 Radiative Flux Divergence for Mars Return Case with Tangent Slab Transport : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
89
6.12 Radiative Flux Divergence for Mars Return Case with MDA Transport 89 6.13 Wall Radiative Flux for AFE : : : : : : : : : : : : : : : : : : : : : :
91
6.14 Radiative Flux Divergence for AFE with Tangent Slab Transport : :
93
6.15 Radiative Flux Divergence for AFE with MDA Transport : : : : : : :
93
6.16 Spectral Distribution of AFE Wall Radiative Flux : : : : : : : : : : :
94
7.1 E ect of Td Models on Molecular Dissociation - 1631 sec : : : : : : : 100 7.2 E ect of Td Models on Temperature Pro les - 1631 sec : : : : : : : : 101 7.3 E ect of Td Models on Radiative Emission Pro les - 1631 sec : : : : : 101 7.4 E ect of v Models on Temperature Pro les - 1631 sec : : : : : : : : 103 7.5 E ect of v Models on Radiative Emission Pro les - 1631 sec : : : : : 104 7.6 E ect of Ev Models on Temperature Pro les - 1634 sec : : : : : : : 106 7.7 E ect of Ev Models on Radiative Emission Pro les - 1634 sec : : : 107 7.8 Predicted Radiation Spectrum - 1631 sec : : : : : : : : : : : : : : : : 110 7.9 Measured Radiation Spectrum - 1631.3 sec : : : : : : : : : : : : : : : 110 7.10 Predicted Radiation Spectra - 1634 sec : : : : : : : : : : : : : : : : : 111
x 7.11 Measured Radiation Spectrum - 1634.43 sec : : : : : : : : : : : : : : 111 7.12 Predicted Radiation Spectra - 1637.5 sec : : : : : : : : : : : : : : : : 113 7.13 Measured Radiation Spectrum - 1636.43 sec : : : : : : : : : : : : : : 113 7.14 E ect of Chemical Kinetics Model on Temperature Pro les - 1634 sec 114 B.1 Variation of and j for a Mars Return Flow eld : : : : : : : : : : 139 B.2 Essential Features of Axisymmetric Radiation Grid : : : : : : : : : : 148 B.3 Top View of Axisymmetric Radiation Grid : : : : : : : : : : : : : : : 149 B.4 Typical Convergence History for MDA Solution : : : : : : : : : : : : 157
xi
List of Tables 5.1 Computational Optimization - FIRE II at 1631 sec : : : : : : : : : :
69
5.2 Computational Optimization - FIRE II at 1634 sec : : : : : : : : : :
69
5.3 Computational Optimization - FIRE II at 1637.5 sec : : : : : : : : :
69
5.4 E ect of Radiation Subgrid Algorithm : : : : : : : : : : : : : : : : :
72
6.1 Selected FIRE II Flight Conditions : : : : : : : : : : : : : : : : : : :
80
6.2 Radiative Heat Flux Predictions for FIRE II : : : : : : : : : : : : : :
85
7.1 E ect of Energy Exchange Models for FIRE II at 1631 sec : : : : : : 116 7.2 E ect of Energy Exchange Models for FIRE II at 1634 sec : : : : : : 116 7.3 E ect of Energy Exchange Models for FIRE II at 1637.5 sec : : : : : 117
Nomenclature a
Grid cell surface area (cm2)
a0
Bohr radius (cm)
A
Transition probability (/sec)
AFE
Aeroassist Flight Experiment
ASTV
Aeroassist Space Transfer Vehicle
b
Normalized line shape (sec)
Blu
Einstein coecient for absorption
B
Rotational constant (eV)
B
Radiosity (W/cm2-sec?1)
c
Speed of light (cm/sec)
CFD
Computational Fluid Dynamics
d
Coecients in Gauss-Seidel formulation
Dv
2nd rotational constant (eV)
Del
Matrix element (statcoul-cm)
D
Dissociation energy from ground state (eV)
DSMC
Direct Simulation Monte Carlo
e
Electron charge (statcoul)
E
Energy or energy level (eV)
f
Distribution function
flu
Oscillator strength for line transitions
xii
xiii
F
Normalized line shape
g
Degeneracy
G
Incident spectral radiative intensity (W/cm2-sec?1)
h
Planck's constant (eV-sec or erg-sec)
h
Energy (eV)
I
Ionization potential of ground state (eV)
I
Radiative intensity (W/cm2-sec?1-ster)
je
Emission coecient (W/cm3-sec?1-ster)
Je
Total line emission
J
Rotational quantum number
J^
Jacobian for transformation from physical to computational coordinates
k
Boltzmann's constant (eV/K or erg/K)
L
Local error in Gauss-Seidel iteration
LAURA
Langley Aerothermodynamic Upwind Relaxation Algorithm
LORAN
Langley Optimized RAdiative Nonequilibrium
m
Mass (g)
MDA
Modi ed Di erential Approximation
n
Principal quantum number
n^
Surface unit normal
N
Number density (cm?3)
NASA
National Aeronautics and Space Administration
xiv NBS
National Bureau of Standards
p
Rotational transition probability
p^
Scattering phase function
q
Franck-Condon factor
~qR
Radiative ux (W/cm2)
Q
Partition function
QSS
Quasi-steady-state
r
Relaxation parameter
~r
Position vector
RIFSP
Radiating Inviscid Flow Stagnation Point
s
Path variable (cm)
S
Radiative source function (W/cm2-sec?1-ster)
t
Time (sec)
T
Temperature (K)
v
Speed (cm/sec)
v
Vibrational quantum number
V
Volume of grid cell (cm3)
W
Weighting factor in subgrid algorithm
x
Cartesian coordinate (cm)
xe
Spectroscopic constant (eV)
y
Cartesian coordinate (cm)
xv
ye
Spectroscopic constant (eV)
z
Cartesian coordinate (cm)
ze
Spectroscopic constant (eV)
Z
Ion charge number
Zk
Radiation subgrid weighting function
Correction to rotational constant (eV)
Correction to 2nd rotational constant (eV)
Line (half) half width (1/sec)
s
Scattering coecient (1/cm)
?
Computational variable for incident intensity
r
Normal distance from wall to center of rst grid cell (cm)
"
Surface emissivity
Third computational coordinate direction
Second computational coordinate direction
Angle subtended by axisymmetric grid (radians)
Absorption coecient (1/cm)
0
Absorption coecient corrected for induced emission (1/cm)
Wavelength (cm)
Frequency (1/sec)
First computational coordinate direction
Cross-section (cm2)
xvi
Molecular state designator
Optical variable
v
Vibrational relaxation time (sec)
!
Solid angle (radians)
!e
Spectroscopic constant (eV)
^
Direction vector
Subscripts: a
perturbing atoms
A
lower electronic state
B
upper electronic state
c
capture
CL
line center
d
dissociation
e
electron
el
electronic
ex
excitation
f
freestream cell face
F
freestream cell center
i
grid index in x-direction
I
cell center index in x-direction
j
grid index in y-direction
xvii
J
cell center index in y-direction
J u, J l
rotational quantum number
k
grid index in z-direction
K
cell center index in z-direction
l
lower state of radiative transition
m
medium
n
index of electronic level
p
Planck
P
J = ?1 rotational transition
Q
J = 0 rotational transition
r
rotational
R
J = 1 rotational transition
R
radiative
s
species index
t
translational
tot
total
u
upper state of radiative transition
v
vibrational
w
wall
x
x component or derivative
y
y component or derivative
xviii
z
z component or derivative
0
reference
frequency
+
ion
Superscripts: B
upper electronic state
D
Doppler
e
emission
J
rotational quantum number
MW
Millikan-White
n
index of time step in iteration
P
Park
R
resonance
s
scattering
S
Stark
v
vibrational quantum number
photoionization threshold
0
integration dummy variable
?
negative y direction
+
positive y direction
1
1 Introduction A signi cant portion of the heating experienced by a blunt spacecraft encountering a planetary atmosphere at high speed can be due to radiation. The radiative heating can therefore be a strong driver on the design of a spacecraft heatshield. In the past, most spacecraft encountering planetary atmospheres were on entry trajectories descending rapidly into the dense lower layers of the atmosphere. The heat load they experienced could therefore be accurately calculated using equilibrium methods. Mission scenarios currently under study, such as Aeroassist Space Transfer Vehicles (ASTV) designed to rendez-vous with the International Space Station Freedom, and aerobrakes used for orbit insertion in planetary exploration, present a di erent situation. Walberg [1] provides a summary of such missions. These vehicles use the upper atmosphere to obtain the necessary velocity change for orbit transfer and may spend signi cant amounts of time in low density regions where nonequilibrium conditions prevail. Numerous Computational Fluid Dynamics (CFD) codes are being developed to solve the Navier Stokes equations for this nonequilibrium regime. These codes predict the convective heating rates for ASTVs. A few studies of the nonequilibrium radiative heating problem have also been made [2, 3, 4]. Nelson [5] discusses some of the problems and preliminary results from nonequilibrium radiative heating analyses. However there remains a need for a relatively fast and accurate method which can
2 be coupled to a ow eld solution in cases of heavy radiation, or used uncoupled in cases with less signi cant radiation. There has also been some evidence in recent years [4] that the one-dimensional tangent slab approximation, which is standard in computing radiative transport, may not be adequate for some of the vehicle shapes and applications under consideration. It will certainly not suce to compute radiative transport in the wake region of such vehicles where payloads will generally be located. The objective of the present work is to develop an accurate, ecient method for calculating radiative heat transfer under nonequilibrium conditions, to initiate the development of a multi-dimensional transport algorithm for such radiative transfer, and to present some initial results. To develop a method for calculating nonequilibrium radiation, the necessary radiation property models were rst derived for a nonequilibrium gas. In nonequilibrium, expressions for both absorption and emission are required since the equilibrium relationship between the two does not apply. This theoretical development is presented in Ch. 3. These nonequilibrium radiation models were then implemented in a computer code dubbed LORAN: the Langley Optimized RAdiative Nonequilibrium code. As its name suggests, this code is optimized to provide radiation predictions eciently, by a judicious selection of spectral points determined from the distribution of radiative transitions. Parametric studies were then performed to determine the minimum set of optimized spectral points which would provide accurate radiative heat ux predictions. This process and its results are described in Ch. 5. The optimized method
3 is compared to ight- and ground-based measurements, and to other nonequilibrium radiation methods, in Ch. 6. The agreement is generally good. To compute radiation transport without applying the tangent slab (1-D) approximation, a full 3-D modi ed di erential approximation (MDA) was adopted. The derivation of this method in nonequilibrium is summarized in Ch. 4, while the numerical details of its implementation are presented in App. B. This method is currently limited to axisymmetric ow elds, though the extension to 3-D ows should be straightforward. The method has been demonstrated for two nonequilibrium ight conditions and compared to results from the tangent slab approximation in Ch. 6. The predicted wall radiative heating is 20 to 25 percent lower than the tangent slab result, in line with what would be expected. There are, however, indications that the boundary conditions of the MDA method can be improved. The radiative coupling term, 5 ~qR, varies much more smoothly in the MDA solution than in the tangent slab prediction, suggesting that the MDA method may enhance the stability of coupled solutions. Chapter 7 applies the LORAN radiation model to a study of the sensitivity of radiative heating predictions. Semi-empirical models for the rate controlling temperature in dissociation reactions, the cross section for vibrational to translational energy exchange, and the amount of energy exchange in dissociation, as well as various chemical kinetics models used in nonequilibrium CFD were varied and the e ect on the predicted radiative heating levels was examined. The results show that radiation is
4 very sensitive to these models through their in uence on the vibrational temperature. Studies of radiative heating may therefore provide additional insight into the validity of the semi-empirical models which have been developed for nonequilibrium CFD codes. The nonequilibrium radiation and transport methods developed for this work predict radiative heating to aeroassist vehicles with accuracy similar to that of detailed codes. The computer time and storage requirements are signi cantly reduced, however. The LORAN method is therefore suitable for use in parametric studies and for coupled solutions with nonequilibrium Navier Stokes codes. Further work on these models is expected to speed up solutions still more.
5
2 Historical Review
2.1 Topical and Literature Survey The previous work which has relevance to this problem can be subdivided into
several subject areas: nonequilibrium absorption and emission in individual radiating mechanisms, nonequilibrium excitation processes, and transport models.
2.1.1 Nonequilibrium Absorption and Emission Emission and absorption of radiation in a gas result from transitions between energy levels. In atomic species, all electronic energy levels as well as free electrons may be involved in radiative transitions. For molecules, radiation also occurs due to transitions between rotational and vibrational energy levels. Atomic radiation arises from bound-bound transitions (atomic lines), bound-free transitions (photoionization or radiative deionization), and free-free transitions (Bremsstrahlung radiation). Here \bound" and \free" refer to the state of the electrons involved in the radiationinducing transition. Molecular radiation is more complex because of the large number of energy levels available for transitions. The resulting structure of closely-spaced lines is referred to as a molecular band system. The complete spectral variation of emission and absorption is contained in the absorption coecient, , and the emission coecient, je . The absorption coecient is obtained from the product of the population, Nn, of an energy level and its radiative
6 absorption cross section, n, summed over all the energy levels available:
=
X energy levels n
Nn n
(2:1)
The emission coecient is found from a similar expression, in which the amount of energy emitted, hnn0 also appears, and the absorption cross section is replaced by the transition probability for emission, Ann0 :
Je =
X energy levels n
Nn
X n0 0, the right-hand-side of this equation must also be positive. This implies for Bvu > Bvl that vu vl > , and for Bvu < Bvl that vu vl < . These requirements are the mathematical statement of the fact that each vibrational band
31 absorbs in only a portion of the spectrum. In evaluating the absorption coecient for a particular band, the wavelength or frequency under consideration must be checked against the above conditions to avoid adding super uous contributions. Returning now to Eq. 3.11 for the rotational energy levels, the assumption of
J l 1 is again made. This allows the replacements J l + 1 J l and 2J l + 1 2J l. Equation 3.38 can then be substituted in Eq. 3.11, with the result: ! ( 1 1 1 1 l l l Er ' hc Be ? e v + 2 ? u l B u ? Bv9l v v v ! 3 2 l = ? 4!Bl2e 1 ? 1u l (B u ?1 B )2 ; v v v e vl
(3.39)
This expression can then be substituted in the rotational exponent in Eq. 3.31. It constitutes the nal approximation of the \smeared band" model. The last term to evaluate in Eq. 3.30 is the averaged frequency, Bvu; Avl . This must be evaluated separately for and non- transitions according to the rotational selection rules discussed above. For a given vibrational transition, Eqs. 3.17 to 3.19 describe the line centers of each rotational transition (with Eq. 3.18 not allowed for transitions). Recalling also that the 0-0 rotational transition is always forbidden,
Bvu; Avl can be evaluated as follows.
3 2Jmax ! l X c 1 1 c 1 1 Bvu ;Avl = ( ) u l = 3J l + 1 4 + + + j l 5 Bv ; Av Q R R J =0 max J l =1 P 3 2Jmax l X 1 1 c 1 c + + j l 5 Bvu; Avl = ( ) u l = 2J l + 1 4 Bv ;Av P R R J =0 l max J =1
where the rst form is for non- transitions, and the second for transitions.
(3:40) (3:41)
32 Substituting Eqs. 3.17 to 3.19 in Eq. 3.40 and 3.41 and collecting terms results in the following expressions for the averaged frequency. l + 1) c u ? 4Du ) + (2 B Bvu; Avl = cu l + c3(JJmax v e l l 3Jmax + 1 v v max + 1 ( l l J 2 max l l l u l 3De ? 3De (Jmax + 1)(2Jmax + 1)(3Jmax + 3Jmax + 1) 30 l 2 l + 1)2 u l ? 15Du + 3Dl + 6Del ? 6Deu Jmax (Jmax + 3 B ? 3 B v v e e 4 l (J l + 1) ) l (J l + 1)(2J l + 1) Jmax Jmax max max max u l u + 3Bv ? 3Bv ? 12De 6 2 l + 1) c u ? 4Du ) + (2 B Bvu; Avl = cu l + c2(JJmax v e l l +1 2Jmax v v max + 1 ( l l J 2 max l l l u l 2De ? 2De (Jmax + 1)(2Jmax + 1)(3Jmax + 3Jmax + 1) 30 l 2 (J l + 1)2 l Jmax max u u ? 2B l ? 14Du + 2Dl + 4De ? 4De + 2 B v v e e 4 l (J l + 1) ) l (J l + 1)(2J l + 1) Jmax Jmax max max max u l u + 2 B ? 2 B ? 12 D v v e 6 2
(3.42)
(3.43)
where again the rst form is for non- transitions, and the second for transitions. While these equations could be implemented as is, it is simpler and still accurate to again neglect the De terms compared to the Bv terms. The result is: # " u (J l + 1) 3(B u ? B l ) 2 B 1 v max v v l l l Bvu; Avl = c u l + 3J l + 1 + 3J l + 1 Jmax(Jmax + 1)(Jmax + 2) v v
max
max
(3:44) u (J l + 1) 2(B u ? B l ) l + 2) # 2 B ( J 1 v max v v max l l Bvu; Avl = c u l + 2J l + 1 + 2J l + 1 Jmax(Jmax + 1) 3 v v max max (3:45)
"
These equations for non- and transitions can then be substituted in Eq. 3.30, and the averaged spectral absorption coecient can be calculated.
33 The nal result for the \smeared band" molecular absorption coecient is
1 X
1 1 3 2 XX q u l u l AB = 83hc ggB DelBA v v Bv ; Av A
vl vu
2J l + 1 v) NA exp (Q?E(Tvl =kT 2cJ ljBvu ? Bvl jQr(Tr ) v v) l J ! ( 1 1 1 1 l l l exp(?hc Be ? e v + 2 ? u l B u ? Bv l v v v 9 ! 3 2 l = ? 4!Bl2e 1 ? 1u l (B u ?1 B l )2 ; =kTr) v v e v v
(3.46)
where is given by Eq. 3.44 for non- transitions, and by Eq. 3.45 for transitions, and the vibrational energy Evl is evaluated from Eq. 3.10 with the appropriate values for the lower energy level. To be consistent, the approximation 2J l + 1 2J l is used to reduce the fraction inside the summation over J l.
Maximum Rotational Quantum Number The maximum rotational quantum number Jmax can be determined to varying levels of approximation. The rst order calculation would predict the maximum to occur when the energy of a level exceeds the dissociation energy of the species. As discussed in Herzberg [65], however, predissociation may occur due to the shape of the energy potential curve. Whiting et al. [19] developed a computer code based on this fact to predict the maximum rotational quantum number. The method was later re ned by Whiting [68]. This most recent method has been adopted for use here. For each vibrational band of each molecular contributor, it computes the maximum allowable rotational quantum number to be considered, based on the shape of the
34 Morse-centrifugal potential describing the rotating and vibrating molecule. Further enhancements to the theory for the maximum rotational quantum number are possible, but it is felt that this level of accuracy is sucient for the current work.
3.2 Emission Under equilibrium conditions, the emission coecient is found from the absorption coecient using Kirchho 's law.
je (s) = (s)Ip(T )(1 ? exp (?h=kT ))
(3:47)
where the factor in parentheses accounts for induced emission and the black body or Planck intensity is
h 3 Ip(T ) = c2 [exp (2h=kT ) ? 1]
(3:48)
Combining these two equations gives the following relation for emission:
je (s) = (s) 2h c2 exp (?h=kT ) 3
(3:49)
which depends on the existence of an equilibrium temperature T . In nonequilibrium, the single temperature T is not de ned. Two approaches for nding the emission coecient are possible. A quasi-equilibrium or excitation temperature appropriate to each radiative transition can be calculated, from which individual contributions to the emission can be obtained. This is the approach adopted in the NEQAIR code [20]. Alternately, expressions for the emission coecient can be found for nonequilibrium from considerations of detailed balancing. In principle,
35 these two approaches should lead to the same result. However, the de nition of an excitation temperature for use in the rst approach is very sensitive to predicted nonequilibrium energy level populations that depend on rate data with signi cant uncertainties. An excitation temperature is de ned by assuming that a Boltzmann or Saha equilibrium exists between the upper and lower level populations for a transition. For some nonequilibrium population distributions negative excitation temperatures can be computed. For bound-free transitions, for example, solving Saha's equation for T gives:
I ? En Tex = k ln (N =N n + =Ne =f (Te ))
(3:50)
If the nonequilibrium population of level n is small the argument of the logarithm may be less than one (the function f (Te ) in this equation is a combination of partition functions), resulting in a negative excitation temperature. Such negative temperatures have no physical meaning and therefore require that some justi cation be developed if they are to be used. The detailed balancing approach avoids this diculty and is therefore employed in LORAN. The necessary expressions for the emission coecients of the various radiative transitions are developed below.
3.2.1 Atomic Mechanisms Bound-Bound Transitions The radiative emission from an atomic line is given by: [6]
J e = Nu hul Aul
(3:51)
36 where Aul is the Einstein coecient for spontaneous emission of the transition from the upper atomic energy level u to the lower level l. Detailed balancing relates Aul to the absorption oscillator strength flu as follows:
2e2 gl 2 f Alu = 8mc 3 g ul lu u
(3:52)
Combining these two equations leads to the spectral emission coecient for boundbound atomic transitions. The line shape b is the same as that for absorption.
he2 gl 3 f N b(; N ; T ; T : : :) je = 2mc e t e 3 gu ul lu u
(3:53)
Bound-Free Transitions Bound-free emission results from the capture of a free electron into an ion. The cross section for this process is given by Zel'dovich and Raizer: [6] 4 4 10 cn = 128p mc3Zh4ev2n3 3 3
(3:54)
where v is the initial speed of the captured electron. The number nc of such captures into the nth energy level for electrons with speeds between v and v + dv per unit volume per unit time is
nc = N+ Nef (v)dv v cn
(3:55)
The electron speeds are distributed according to the distribution function f (v) which is assumed to be a Maxwell distribution in equilibrium at the electron translational temperature Te. The energy emitted, hul, where the subscript u now denotes the
37 free state, is given by the sum of the initial kinetic energy of the electron and the net energy required to reionize it from the nal energy level l.
hul = m2v + (I ? El) 2
(3:56)
I is the ground state ionization potential. El is the energy of the bound level, measured from the ground atomic state. This equation is solved for v2 and substituted to eliminate v in favor of the frequency in Eqs. 3.54 and 3.55. Combining all these pieces, the emission coecient for bound-free transitions is found from:
je d = The complete expression is
X
1 h n d ul c nal levels l 4
(3:57)
m 3=2 nX 4 Z 4e10 max 1 h ? I + El ) 128 exp ? ( (3:58) = p m2c3h2 N+ Ne 2kT 3 kTe 3 3 e nmin n The lower limit on the summation corresponds to capture of a zero energy electron. je
nmin is determined for each frequency by setting v to zero in Eq. 3.56. The upper limit is determined by ionization of the atom.
Free-Free Transitions Free-free transitions are a special case in nonequilibrium because they involve only free electrons. Since electrons are assumed to equilibrate rapidly to some temperature
Te, Kirchho 's law and the development of the emission coecient lead to the same result.
2 1=2 Z 2e6 je = 38 3mkT mc3 N+Ne exp(?h=kTe) e
(3:59)
38
3.2.2 Molecular Mechanisms A \smeared band" expression can be developed for the molecular emission coecient by a process parallel to that for absorption. The total emission in a rotational line, Jule , is obtained from the Einstein coecient for spontaneous emission:
Jule = Nu Aulhul
(3:60)
The emission intensity at a particular frequency is obtained by dividing by the solid angle and introducing the line shape function, F ( ).
jeul = Nu A4ulhul F ( )
(3:61)
Substituting Eq. 3.20 for the Einstein coecient, this becomes 2 qvu vl pJ u J l F ( ) jeul = Nu 163c2 ul4 DelBA 3
(3:62)
To obtain the \smeared band" result, this expression must rst be summed over J l and the average frequency Bvu; Avl introduced. Using the summation rule for pJ u J l this gives
4 u l D2 jeAvl; Bvu Ju = NBvuJ u 163c2 Bv ; Av elBA qvu vl F ( ) 3
(3:63)
Proceeding in parallel to the development for absorption, the sum over vl is now carried out.
2 jeA; BvuJu = NBvu J u 163c2 DelBA F ( ) 3
1 X vl
4 u l qvuvl Bv ; Av
(3:64)
where again the frequency remains inside this summation for accuracy. The average emission coecient for a single rotational line is now obtained using the mean value
39 theorem. Z
jeA;Bvu Ju d line
1 3 X 16 2 4 u l =je = 3c2 DelBA NBvuJ u qvu vl Bv A; Bvu J u ; Av vl
(3:65)
where is the width of the line. This equation can now be solved for the rotationally averaged emission coecient. The complete \smeared band" emission coecient is then obtained by carrying out the summations over the nal rotational and vibrational levels, J u and vu. The result is:
1 N u u X Bv J (3:66) vu vl J u As in the case for absorption, the in nite sums are truncated by the occurrence of 2 j eAB = 163c2 DelBA 3
1 1X X
4 u l qvuvl Bv ;Av
dissociation. The average frequency, Bvu; Avl , in Eq. 3.66 is the same as that for absorption, and is given by Eqs. 3.44 and 3.45 for non- and transitions, respectively. The average line spacing is also the same, and is given by the expression in Eq. 3.37 with the rotational quantum number expressed in terms of the wave number according to Eq. 3.38. The nal term in the emission coecient, the nonequilibrium upper level population, is found from the expression: u v ) gJ u exp (?EJ u =kTr ) NBvuJ u = NB exp(Q?E(Tv =kT ) Q (T )
v v
r r
(3:67)
where the vibrational and rotational energies are given as a function of wave number by Eqs. 3.10 and 3.39, evaluated for the upper energy level, and the electronic level population, NB , is obtained from the QSS method. The evaluation of the partition functions was discussed in the section on absorption.
40 Substituting all these expressions in the emission coecient, Eq. 3.66, the \smeared band" result is obtained: 2 j eAB = 163c2 DelBA 3
1 X
1X 1 X vu vl
4 u l qvuvl Bv ; Av
u 2J u + 1 v) NB exp (Q?E(Tv =kT 2cJ u jBvu ? Bvl jQr(Tr ) v v) u J ( ! 1 1 1 1 u u u exp(?hc Be ? e v + 2 ? u l B u ? v 9 Bv l v v ! 3 2 u = ? 4!Bue2 1 ? 1u l (B u ?1 B l )2 ; =kTr ) v v v e v
(3.68)
Again the approximation 2J u + 1 2J u is used inside the summation over J u, to be consistent with the level of accuracy of the result.
3.3 Induced Emission The above expressions for the emission coecient do not include all the radiation emitted by the medium. Stimulated or induced emission also occurs due to the presence of photons. This emission is proportional to the radiative intensity, I , and is therefore commonly included as a correction to the absorption coecient. The induced emission is given by:
j0
2 c e = j 2h 3 I
(3:69)
In equilibrium the intensity, I , in this equation becomes the Planck black body function, and the result of Eq. 3.47 is obtained. The absorption coecient corrected for induced emission can then be identi ed in equilibrium from Eq. 3.47 as
0 = (s)(1 ? exp (?h=kT ))
(3:70)
41 In nonequilibrium, the intensity cannot be replaced with the Planck function. Instead, the induced emission coecient of Eq. 3.69 must be substituted in the transport equation, along with the coecients for spontaneous emission and induced absorption. The transport equation, Eq. 2.4, then becomes:
@I (s; ^ ; t) + (s)I (s; ^ ; t) = j e (s; t) + j 0 (s; t) @s
(3:71)
where je is the spontaneous emission coecient. Substituting Eq. 3.69 and gathering terms containing I then allows the de nition of a corrected absorption coecient equal to the factor multiplying I . The result is:
c2 0 = ? je 2h 3
(3:72)
The corrected absorption coecient may be negative in nonequilibrium. This fact must be carefully considered when developing any solution algorithm for radiative transport in nonequilibrium problems. Negative values of 0 occur when the nonequilibrium populations are such that the second term on the right-hand-side of Eq. 3.72 is larger than the rst term. Physically, this means that the upper energy level of a transition or transitions is overpopulated relative to the lower energy level. Such a population inversion can occur in a nonequilibrium boundary layer, for instance, when the higher energy levels are still populated according to the temperature and chemistry of the inviscid shock layer. Because of the ?3 term appearing in Eq. 3.72, negative values of 0 tend to occur at the low energy end of the spectrum (see Fig. 5.3).
42
4 Transport Theory Solving the equation of radiative transport (Eq. 2.4) in a tractable computation requires a number of approximations. It is not possible to consider an in nite number of frequencies or an in nite number of directions. Assumptions commonly made are a step-wise variation of radiation properties over part or all of the frequency spectrum, and a one-dimensional variation of the gas properties. The rst assumption may be acceptable in many situations, but must be used cautiously in nonequilibrium. The absorption coecient in nonequilibrium cannot be found simply from temperature and equilibrium composition but depends on the population of excited states, and hence the chemical and excitation kinetics. This introduces a number of new variables and makes the design of a step absorption coecient dicult. Some method of selecting appropriate frequencies at which to compute the radiative properties must be devised. This problem will be discussed in Ch. 5. The second assumption of one-dimensional property variation should be nearly valid in the stagnation region of ows over blunt bodies. However, the results shown in Figs. 6b and 11 of Candler [4] suggest that even near the stagnation point this approximation may be in error. The assumption certainly introduces considerable error in the radiative ux at corners, and is not at all valid in the wake region of vehicles such as the proposed ASTV. Such wake ows are of great interest, because of
43 the desire to place unshielded payloads in the lee of these vehicles for orbit transfer.
4.1 Flow eld Coupling Including gas radiation e ects in a thermochemical nonequilibrium ow eld calculation means including the coupling between the uid dynamics and the radiation. The coupling occurs because of the radiative ux term 5 ~qR (denoted Qrad by Gno o [69]) which appears in the total and vibrational-electronic energy conservation equations. In the tangent slab approximation, the divergence of the radiative ux reduces to simply the normal derivative of qR. The appropriate term is obtained from the radiative transport solution as discussed below for each solution method considered.
4.2 Plane-Parallel Medium A common approximation for solving the radiative heat transfer equation is to assume that the medium through which the radiation travels varies in only one direction. This is often a reasonable assumption in the stagnation region of a ow eld, as well as in other geometries which are of interest for radiant heating. In this situation the path variable s can be replaced by the Cartesian coordinate y using the chain rule:
@ = @ dx + @ dy + @ dz @ dy @s @x ds @y ds @z ds @y ds
(4:1)
44 0.150
Freestream Boundary y or s
0.100
0.050
x
-0.000
z
-0.050
-0.100
Body Surface
-0.50
-0.2 5
0.00
0.25
Figure 4.1. Shock Layer Geometry for Plane-Parallel Approximation For a shock layer geometry, if both s and y are taken along the surface normal direction as shown in Fig. 4.1 then
@ = d @s dy
(4:2)
dI (y; t) + 0 (y)I (y; t) = j e (y; t) dy
(4:3)
Equation 2.4 then becomes
This is the equation of radiative transfer along the y-axis in a plane-parallel medium under conditions of nonequilibrium.
4.2.1 Transparent Gas A transparent gas is one in which the absorption coecient is assumed zero. In this case, Eq. 4.3 can be further simpli ed to obtain:
dI (y; t) = j e (y; t) dy This equation can easily be integrated to nd Zy I (y; t) = je (y; t) dy y0
(4:4)
(4:5)
45 For a solution over a ow eld grid, in which the emission coecient is known at a number of discrete points, this equation can be solved by any numerical quadrature method. For a transparent gas, the radiative ux ~qR can be found simply by multiplying the intensity by 2. The divergence of ~qR, needed for coupling to a ow eld, can be obtained by numerical di erentiation. Despite the assumption that the gas does not absorb radiation, this term will be nonzero due to emission from the gas.
4.2.2 Absorbing Gas For an absorbing gas Eq. 4.3 can be integrated formally to obtain an expression for the radiative intensity for any location y. This has been done in equilibrium for instance by Ozisik [21, p. 267]. It is convenient to split the intensity into two components: one directed along +y, denoted I+, and the other along ?y, denoted
I?. Ozisik's equations can be adapted to the nonequilibrium situation by recognizing that the source function, S , in a non-scattering nonequilibrium gas becomes je=0 . The formal solution for these two components of the radiative intensity can then be written for the nonequilibrium problem as:
Z j e( 0 ) e?( ? 0 ) d 0 0 ( 0 ) 0 Z 0 j e( 0 ) 0 e?( ? ) d 0 I?( ) = I?( 0 )e?( 0 ? ) + 0 ( 0 ) I+( ) = I+(0)e? +
(4:6) (4:7)
where is the optical variable de ned for each frequency as
=
Zy 0
0 (y)dy
(4:8)
46 The = 0 boundary is the vehicle surface, which is assumed to emit radiation according to Planck's function at the wall temperature Tw . A radiation equilibrium wall temperature can be computed for this purpose, but in this analysis Tw is assumed to be given. The = 0 boundary is the freestream gas ahead of the vehicle. The radiative intensity emanating from the freestream is assumed to be zero. The second equation therefore reduces to:
Z 0 j e ( 0 ) 0 ? I ( ) = 0 ( 0 ) e?( ? )d 0
(4:9)
Nicolet [24] assumed a log-linear variation of properties to solve these equations. This formulation does not accommodate negative values of 0 , the absorption coef cient corrected for induced emission, which can occur in nonequilibrium. Nicolet's transport algorithm can therefore not be used here. As in the transparent gas case, any numerical quadrature method can be used to obtain the radiative intensity distribution along the body normal, but it must allow for negative 0 . To obtain the radiative ux in an absorbing, plane parallel medium it is easy to show that the above equations apply with the simple replacement of by 2 and the addition of a factor of . The net radiative ux is then given by
( ) Z j e ( 0 ) Z 0 j e ( 0 ) 0 ) 0 ? ) + ? 2 ? 2( ? 0 ? 2( 0 qR ( ) = I (0)e + 0 ( 0 ) e d(2 ) ? e d(2 ) 0 0 ( 0 )
(4:10)
The divergence of the radiative ux is obtained by di erentiating this equation with
47 respect to y, then integrating over the complete spectral range. The result is:
( " #) @qR = Z max ?20 ( ) I +(0)e?2 ? 2 je ( ) + Z je ( 0 ) e?2j ? 0 jd(2 0 ) d @y min 0 ( ) 0 0 ( 0 ) (4:11) where is a function of y. For a solution including all the radiative energy, min is zero and max is in nity. In practice, nite limits on the spectral integration will be required. Most gas radiation in shock layers in air occurs at energies below 16.5 eV. An alternate upper limit at 6.2 eV includes only the visible and infrared portions of the spectrum. This range of energies is easier to measure experimentally and includes minimal self-absorption. The lower limit is often set around 0.3 eV because a zero energy photon causes a singularity in the free-free absorption coecient (Eq. 3.9). The singularity arises because additional quantum mechanical considerations must be included at very low energies. The radiative energy omitted by ignoring these low energies is minimal.
4.3 Multi-Dimensional Medium Since an exact solution of the radiative transport equations in multi-dimensional media is not feasible, an approximate method is sought. Such a method should introduce considerable simpli cations to the governing equations in keeping with the stated objective of providing a relatively fast solution. Numerous approximate methods are available in the literature. A brief summary was provided in the topical review in Ch. 2. The best-developed of these methods are the moment methods, which reduce
48 the problem from a solution of integro-di erential equations to a solution of simpler ordinary di erential equations. Many of these methods have been developed speci cally for one-dimensional radiative transport. Sparrow and Cess [70, Sec. 7-9] caution that the extension to three dimensions is open to question and must be made with caution. They also show, however, that the moment method reduces to the correct expressions in the limit of optically thin or optically thick gases and suggest that it should therefore be of reasonable accuracy for all values of optical thickness. Modest has modi ed the di erential approximation to make it even more accurate for small optical depths and generalized it for the three-dimensional case. His method gives results of excellent accuracy for all optical conditions [27]. It is adapted below for the current non-gray, nonscattering, and nonequilibrium ow. As previously mentioned, the source function in a nonequilibrium, nonscattering medium is
e S = j0
(4:12)
The de nitions of the incident intensity and ux are repeated here for convenience: Z G (~r) = I (~r; ^ )d! (4:13) 4 Z (4:14) ~qR (~r) = I (~r; ^ ) ^ d! 4
In what follows the R subscript on ~qR will be dropped. The intensity is now divided into two separate contributions: one due to the emission from the medium, Im; and the other traceable to the wall emission, Iw: I (~r; ^ ) = Iw (~r; ^ ) + Im(~r; ^ ). The incident intensity and ux will then also have two components: G = Gw + Gm and
49
~q = ~qw + ~qm. While the intensity attributable to the wall may be highly directional, that emanating in the medium should vary quite smoothly. This portion is therefore assumed to be given by the P-1 approximation, which is the simplest form of the moment method. In this case, the medium intensity can be expressed as
Im 41 (Gm + 3~qm ^ )
(4:15)
This expression is unchanged from the work of Modest [27]. The equation governing the medium intensity then becomes
5 ( ^ Im ) = je ? 0 Im (~r; ^ )
(4:16)
where the nongray, nonscattering, nonequilibrium nature of the gas has been accounted for. Taking the zeroth and rst moments of this equation and integrating over 4 steradians yields the applicable governing equations for G and ~q of the medium:
5 ~qm = 4je ? 0 Gm
(4:17)
5 Gm = ?30 ~qm
(4:18)
The details of the numerical solution to these equations are given in Appendix B. To obtain the divergence of the radiative heat ux, which is required to couple the solution with the ow eld, the contribution to ~q from the walls must also be considered. This is obtained from the following integral:
Z 1 ~qw = B [~r0(!)]e? ^ d! 4
(4:19)
50 where the wall radiosity, B , is given by the sum of emission at the wall temperature
Tw , and re ection of radiation emanating from other wall surfaces. In Modest's analysis, the wall re ection is assumed to be di use. The treatment of specularly re ecting wall surfaces adds considerably to the complexity of the problem and will not be addressed here. In any case, for a forebody ow eld the wall surface is convex so that no re ection of radiation emitted by the wall can occur. Then the radiosity is simply the emission from the wall:
B (~r) = " Ipw (~r)
(4:20)
This contribution to the radiative ux must be computed for each grid cell in the medium by integration over all the wall surface elements to which it has a line of sight. If it is further assumed that the wall is cold, or that it makes a negligible contribution, then ~qw 0, and only the medium contribution need be considered. This approximation is not unreasonable for aeroassist ow elds, since heating rates are relatively low in the upper atmosphere and the wall temperature will accordingly be relatively low during much of the ight. In fact, at the maximum wall temperature predicted for the Aeroassist Flight Experiment (AFE), the peak of the black body emission spectrum occurs at about 0.75 eV. This temperature is close to the limit achievable for reusable surfaces using Space Shuttle-type thermal protection system tiles, and so is representative of the maximum wall temperature to be expected for a non-ablating surface. For lower temperatures, the peak shifts to even lower energies. In this low energy spectral region the gas is quite transparent to radiation (see
51 Fig. B.1) so the wall can have little e ect on the divergence of the radiative ux. In that case, 5 ~qR 5 q~m and is obtained by integrating Eq. 4.17 over frequency. Note that unlike the plane-parallel medium, numerical di erentiation is not required here. Boundary conditions in the multi-dimensional case are less obvious than for the plane parallel approximation. It may again be assumed that the radiative intensity emitted in the freestream is zero. Even in a three-dimensional ow eld, however, symmetries are generally exploited. This means that some grid boundaries will be symmetry planes. Others will be out ow boundaries. Symmetry planes can be treated with re ecting boundary conditions, but the treatment of out ow boundaries is illde ned. The conditions of the gas are not known beyond such boundaries, though it is expected to continue radiating for some nite distance for most ight conditions of interest. The treatment of out ow boundaries is discussed further in Sec. 6.2.1 and Appendix B. For the medium intensity, the vehicle surface is considered to be a cold wall, for which Marshak's boundary condition [21] for the P-1 method is 2 2 " ? 1 ~q n^ + G = 0
(4:21)
The ux, if any, impinging on a wall surface due to emission or re ection from other wall surfaces must be obtained by an integration which considers the view factors of the particular problem. When the wall is convex, this contribution is zero. The numerical details of the treatment of the boundary conditions are given in Appendix B.
52
5 Implementation The implementation of the theory for radiation transitions and radiative transport developed in the previous two chapters in a practical computer code requires some additional work. The details of the process are described in this chapter.
5.1 Coding The present method is implemented in the LORAN (Langley Optimized RAdiative Nonequilibrium) code, which was developed using two existing codes as starting points. The RAD/EQUIL program [33] was followed in developing some of the code structure. It also is the source of part of the algorithm used to model atomic line radiation. The quasi-steady-state (QSS) excitation portion of the NEQAIR program [20] has been minimally adapted for use in LORAN. LORAN is divided into four major sections: reading data and setting up properties which are constant for a calculation, calculating the nonequilibrium excitation, evaluating the radiative properties, and solving the radiative transport problem.
5.1.1 Initial Setup Data Input To evaluate the expressions for absorption and emission obtained in Ch. 3 for each of the radiating mechanisms, input values are required. Data are also required to predict the nonequilibrium populations of the various bound energy levels. Flow eld
53 data are required to de ne the local gas conditions. Finally, program control inputs are desirable to maximize the exibility of the method without requiring constant reprogramming and recompiling. These should allow the user to turn individual radiating mechanisms or species on or o to perform detailed studies of the radiation spectrum. They should also provide for selection of options in the code (i.e. tangent slab vs. 3-D transport). These inputs are obtained from a variety of sources, as discussed below. Most of the inputs are species properties and will not change unless signi cantly di erent new data become available.
Flow eld Data The information required about the gas conditions consists of the number densities of each of the radiating species considered and temperatures for the several energy modes. The Langley Aerothermodynamic Upwind Relaxation Algorithm (LAURA)
ow eld solver [71] is the source of the ow eld data used in developing this method. It is an 11-species model which provides number densities for all the nitrogen/oxygen radiating species of interest. LAURA currently incorporates a two-temperature model for thermal nonequilibrium [72]. The rotational and heavy particle translational energy modes are assumed to be equilibrated, while the vibrational and electron translational energy modes are assumed to be in equilibrium with each other but not necessarily with the other modes. This means that the separate Tv and Te carried in the development of the absorption and emission coecients are equal, as are Tr and
Tt. The distinction between these temperatures was maintained in LORAN, however,
54 so that each can be used if the separate temperatures become available from LAURA or another Computational Fluid Dynamics (CFD) code in the future. Some recent work [73] suggests further that distinct vibrational temperatures may exist for each molecular species. Should a CFD code ever include such detail, minor modi cations to LORAN will allow these individual Tv values to be used. An e ort has been made to keep the input of the ow eld properties from LAURA generic, so that with minor changes to a single subroutine LORAN can be interfaced with a di erent CFD code.
Maximum Rotational Quantum Number The radiative property data discussed above are read into the program rst. As part of this process, the maximum rotational quantum numbers are assessed using the algorithm developed by Whiting (see Sec. 3.1.2). These values do not change in the program.
Continuum Spectrum The spectral location of all radiating transitions is determined from the set of energy levels in the atoms and molecules appearing in the gas. This information can be used to select a minimum set of spectral points for use in the radiation calculation. Consider the bound-free continuum. The absorption coecient for this mechanism is a smooth function of frequency with discontinuities corresponding to the activation of additional energy levels. To resolve this spectrum, spectral points must be
55 1.00
0.90
E/Emax
0.80
0.70
0.60
0.50
0.40 2 .50
2 .75
3.00
3.2 5
3.50
hν, eV
Figure 5.1. Example of Spectrum Optimization for Bound-Free Continuum placed within a very small interval on either side of each jump, as shown in Fig. 5.1. However, this interval is much smaller than what is needed for the smooth portions of the curve. RAD/EQUIL took advantage of this fact by allowing the user to input a few spectral points to capture the jumps. In LORAN, this procedure is automated and incorporated in the radiation calculation. For the particular set of atomic species considered in a computation, a tailored atomic spectrum is generated to resolve the active discontinuities, as the gure illustrates. This simpli es the inputs and eliminates the chances of omitting a level. It also allows the atomic continuum calculation to be performed on the minimum resolving array of spectral points. As discussed in Ch. 3, the molecular radiation has been modeled here using the \smeared band" approximation. In this approximation, the rotational structure is
56 smoothed into an exponential variation. To resolve the resulting spectrum, spectral points must be placed around each vibrational band head. A few points are sucient to capture the smoothed rotational variation within each such band. The resulting structure is similar to the bound-free spectrum, though it is much more complicated because the vibrational bands are far more numerous and often overlap each other. These considerations have been used to develop a routine to choose an optimum molecular spectrum for each input spectral range and set of active species. For each active species, the vibrational band heads that occur in the spectral range under study are computed from Eq. 3.36 with J = 0. An initial spectrum is generated by resolving the band head and spacing nv points in the interval between the zero rotational level (J = 0) and the maximum rotational level (J = Jmax). An example for a major vibrational band of N2+ is given by the curve labeled \initial spectrum" in Fig. 5.2. Note that the clustering of points along this curve results from additional weaker bands which underlie this major band. Since the absorption coecient decreases rapidly for a given vibrational band (note the log scale in the gure), this spectrum can be reduced. This is done by discarding points that describe a vibrational band away from its band head (points required to resolve the jump are kept), if they are within a given interval crit of another point. This procedure can reduce the spectral array by nearly an order of magnitude. A second reduction can be made by discarding any point which has a neighbor within a smaller interval
crit2. Applying these steps to the spectrum in Fig. 5.2 results in the optimized
57 0.0
Log10 (κ ν/κ head)
Optimized spectrum (2 7 points)
-2 .5
-2 .5 Initial Spectrum
-5.0
(88 points) -5.0
-7.5
Log10 (κ ν/κ head)
0.0
-7.5 -10.
-10.0 2 .900
2 .950
3.000
3.050
3.100
3.150
hν, eV
Figure 5.2. Example of Spectrum Optimization for a Vibrational Band spectrum shown. Applying them to the entire set of vibrational bands, a spectrum can be generated that resolves the molecular transitions using less than 2000 points and that is optimized for whatever species and spectral range have been requested. This calculation is computationally expensive, since it sifts through a large number of possible spectral points to produce the optimized molecular spectrum, and it is therefore only exercised once per run. The resulting spectrum is stored for all further computations. Appropriate selection of the parameters nv, crit, and crit2 controls the amount of detail included in the molecular spectrum and its size. Once an optimized spectrum is obtained, the index of the rst and last spectral point for each vibrational band is determined. These indices simplify the logic required in computing the molecular band radiation, and improve its eciency.
58 A few (about 50) points from the optimized molecular spectrum that resolve strong vibrational transitions are identi ed and interleaved into the atomic continuum spectrum discussed above. The resulting spectrum of about 150 points will be referred to as the modi ed atomic spectrum. Its use is discussed in Sec. 5.1.4.
5.1.2 Nonequilibrium Excitation To obtain the nonequilibrium electronic level populations for the atoms and molecules in the ow eld, data on the rates of excitation and deexcitation of each level are required. Sources for this data are varied, but the rates presented are often uncertain due to the experimental diculties involved in their measurement. Park [20] has collected a complete set of rates for nitrogen and oxygen species which has been adopted here. The nonequilibrium excitation calculation in the present method was taken from Park's NEQAIR program. It consists of a QSS solution to the set of excitation rate equations, in which it is assumed that the rate of change in a level's population is small compared to the rate of transitions into and out of that level [67, Chapter 3]. This method is designed for use in situations where the amount of nonequilibrium is \not too large". Its most serious weakness in the present application is that it does not account for the history of the ow in areas of high gradients (boundary layer and shock). The excitation calculation depends on excitation rate data for the various energy levels. This type of data is rather dicult to obtain and therefore has a signi cant
59 level of uncertainty. Park [3] cites this uncertainty as a factor of two, and asserts that the impact on the excited state populations is negligible. Further veri cation of this accuracy would be desirable in the future.
5.1.3 Radiation Properties Bound-Bound Mechanisms For bound-bound transitions the oscillator strengths flu , line centers CL and Stark (half) half-widths S are needed. The National Bureau of Standards (NBS) has published a large number of line centers and oscillator strengths [66], while Stark (half) half-width information is available from Griem [63], among others. In addition, the energy levels and degeneracies of the upper and lower levels of the transitions can be found in the NBS compilation. In the RAD/EQUIL code, Nicolet approximated the so-called high series lines by an integral. These lines have lower state quantum numbers of four or greater, and are numerous and closely spaced at energies below 3.2 eV. Park treats each of these lines individually in NEQAIR. In the present method, in order to reduce the number of lines computed and therefore the run time, the high series lines were represented by multiplet-averaged values. Data for these averaged lines were obtained from the NBS tables, for transitions corresponding to Park's inputs. This approximation should have minimal impact on the results, and yet allows the number of lines treated to be reduced by more than half. The line shape models for LORAN are taken from the development by Nicolet
60 [23] as implemented in the RAD/EQUIL code [33]. This includes natural, Doppler, resonance, and Stark broadening of the atomic lines, as discussed in Ch. 3. Lorentz or Doppler line shapes are selected depending which line width is larger. Each line is resolved by a small number nppl (generally less than or about 15) of spectral points that are distributed starting from the line center (see Fig. 3.1). The location of the spectral point farthest from the line center is determined from the width of each atomic line. In Nicolet's implementation the line width used to distribute these points was an approximation to the maximum width in the gas layer so that all or most of the energy in each line would be included. This approximate maximum width was calculated at a location input by the user at which the maximum temperature was expected. For an application in which a complete ow eld is to be computed, the identi cation of the occurrence of maximum line width presents some diculties. Further, since the line width may vary by orders of magnitude between various points in a
ow eld, the use of a spectrum geared to the largest width can result in signi cant loss of accuracy. For these reasons, the routine which sets up the line frequency spectrum (subroutine freq of RAD/EQUIL) has been modi ed to be applied to every point in the ow eld. This results in the speci cation of a local set of spectral points based on the local line width. These disparate frequency spectra are then reconciled to generate a single atomic line spectrum. This process results in the selection of a set of spectral points capturing the energy in a line at its broadest, but involving
61 detailed spectral information at each grid point. This technique allows satisfactory resolution of the atomic lines, without requiring an excessive amount of storage or computation. An additional change made to Nicolet's method arises because Stark broadening is not always the dominant e ect on line width. In the original RAD/EQUIL, the point distribution logic in subroutine freq considered only the Stark line width. In the present model, the largest of the Doppler and Lorentz (Stark e ect) line widths is used. While this is a small change, it should result in more complete coverage of the atomic line radiation in the present model. The RAD/EQUIL method has also been modi ed to use nonequilibrium excited state populations and electron temperature, where appropriate. The Doppler line width remains a function of heavy particle temperature, since it arises from the thermal motion of the atoms themselves.
Bound-Free Mechanisms Computing the radiation from atomic mechanisms requires knowledge of the energy levels of each atomic species. The large number of electronic energy levels in Eq. 3.8 must be modeled by a manageable set of inputs. To reduce book-keeping and complexity, a common practice is to group neighboring energy levels into one, with an appropriate overall degeneracy. This practice has been adopted here. The grouped levels from Park's QSS method [20] are used for nitrogen and oxygen atoms (twenty-two levels for atomic nitrogen, nineteen levels for atomic oxygen). The re-
62 quired values for the principal quantum number n and the energy level En for each of these levels is input. The bound-free photoionization edge structure is produced by the activation of each level in the summation term of Eq. 3.8. This requires determining the lowest accessible bound energy level, n, for each spectral point in the atomic continuum spectrum. The ground state cross sections are invoked as constants to avoid the errors arising from the hydrogenic model, as discussed in Ch. 3. Each of the atomic energy levels is treated individually in the bound-free calculation, but since these are grouped levels many more energy levels are in fact approximately included.
Free-Free Mechanisms LORAN considers only the radiation produced by an electron slowed by the presence of an atomic ion. The free-free radiation produced by the proximity of a neutral atom or a molecule is considered negligible. The free-free contribution is computed on the same spectrum as the bound-free radiation. The resolution thus obtained is more than sucient, and allows the two atomic continuum processes to be easily combined.
Molecular Mechanisms To predict the molecular band radiation, spectroscopic constants describing the distribution of rotational and vibrational energy levels according to Eqs. 3.11 and 3.10 must be provided. These can be found in a number of sources, such as Herzberg [65], Bond et al. [74], and many others. These sources also provide the necessary informa-
63 tion on the electronic energy levels and energy level degeneracies in molecules. The intensity of radiation depends on the Frank-Condon factors qvu vl and the electronic transition moments DelBA . Data on these quantities may be found in many often contradictory sources. The complete set of molecular inputs collected by Park for his NEQAIR code [20] has been adopted here for simplicity. The molecular band radiation is computed using the optimized molecular spectrum whose determination was discussed in Sec. 5.1.1. The expressions for the emission and absorption coecients developed in Ch. 3 have many terms in common. Advantage is taken of this fact to reduce the computation time required. Since the molecular calculation remains one of the more expensive parts of the radiation calculation, care was taken in programming this particular mechanism to ensure that the advantages of vectorization are exploited to the extent possible.
5.1.4 Radiative Transport Two distinct methods to compute radiative transport were developed in Ch. 4, with and without the tangent slab approximation. Both employ an approximate treatment of atomic lines based on the method of RAD/EQUIL. A single mean value of the continuum (atomic continuum plus molecular band) absorption and emission coecient is computed for each line and added to the detailed absorption and emission coecients calculated at the nppl points describing the line. These mean values are obtained by interpolating the information in the optimized continuum spectra, thus avoiding an expensive computation at each of the nppl spectral points describing
64 each line. Since each atomic line covers only a very small part of the spectrum (see for instance Fig. B.1), this approximation should introduce minimal inaccuracy. Radiative transport is then computed at all nppl spectral points describing each line. This detailed result can be integrated over frequency to provide an average ux for each line if a concise presentation is desired, or it can be presented in full detail. The molecular spectrum can be averaged to obtain values at each point in the much smaller modi ed atomic continuum spectrum, so that the number of spectral points for which transport must be calculated is signi cantly reduced. Since these approximate results are based on a detailed molecular spectrum of both emission and absorption, the prediction of molecular radiation transport should be quite accurate. This averaging technique has been used for the full ow eld solutions reported in Ch. 6 to reduce the computation time. For those cases in which only the stagnation line is computed, the complete optimized molecular spectrum is used in the transport calculation.
Tangent Slab The numerical integration of the transport equation simpli ed for the case of onedimensional radiation (Eq. 4.3) is straightforward, with one exception. The method used by Nicolet in the RAD/EQUIL code assumes a log-linear variation of the absorption coecient between grid points [24]. As mentioned in Sec. 3.3, under nonequilibrium conditions there is no longer any assurance that the absorption coecient corrected for induced emission, 0 , will be positive. The log-linear variation assumed
65 by Nicolet cannot handle a negative 0 . For this work, a simple linear variation between grid points is used instead.
Multi-Dimensional Transport The details of the modi ed di erential approximation used to solve the radiative transport equation for multi-dimensional media are discussed in App. B, along with related numerical considerations. When 0 is not positive, numerical instabilities have been observed in this method. Further work is needed to determine whether these instabilities can be removed. For now, spectral points at which these instabilities occur are ignored. Figure 5.3 is a plot of the number of occurrences of a negative 0 in a test ow eld at each point in the radiation spectrum. It shows that signi cant numbers of negative 0 only occur for a few points in the low energy end of the spectrum. The e ect on predictions of wall radiative ux from ignoring these few spectral points will be quite small (see Ch. 6). It should also be pointed out that the occurrence of these negative values of 0 is controlled by the nonequilibrium excitation calculation. As mentioned before, this calculation contains signi cant uncertainties. An example of a population distribution at a point with negative 0 is shown in Fig. 5.4 (an equilibrium distribution would appear as a straight line in this gure). This population distribution is quite odd, with neighboring energy levels having populations di erent by many orders of magnitude, and may be due to uncertainties in excitation rates. It is possible, therefore, that at least some of the negative 0 values result from uncertainties in the excitation
Number of negative κ ν’ in flowfield
66 12 50
1000
750
500
2 50
0 0
5
10
15
hν, eV
Figure 5.3. Occurrence of Negative 0 in a Test Flow eld calculation. Ignoring these spectral points may therefore introduce less error than including them.
5.2 Computational Optimization Obtaining radiation predictions for complete ow elds, or coupling radiation and
ow eld solutions, requires that the radiation calculations be carried out many times. In order to make this practical and to minimize the cost, it is desirable to optimize the radiation program, and to determine the minimum set of calculations which will provide an accurate radiation prediction. Methods of achieving this are discussed below.
67 15.0 12 .5 10.0
Ei, eV
7.5 5.0
2 .5 0.0 0
5
10
15
Log10 (Ni/gi)
Figure 5.4. Nonequilibrium Population of Excited States of O for Negative 0
5.2.1 Radiation Calculation Parametric studies were conducted to determine the spectral resolution required for acceptable accuracy in the predicted radiation. Variables whose e ect was studied include na, the number of points allowed in the atomic continuum spectrum; nppl, the number of points resolving an atomic line; and the several variables which control the generation of the spectrum for molecular band radiation (nv, crit, crit2). Tables 5.1-5.3 summarize the ndings of this optimization process. The baseline against which the ux calculation is judged ( rst line of each table) is a detailed spectrum case with na =1500, nppl =15, nv =10, crit =0.00825 eV, and crit2 =0.0011 eV. While uncertainties in the radiation calculation mean that this prediction is not necessarily right, such a detailed spectrum does provide the best result possible. The run
68 times are judged against a case with na =100 and all other values unchanged (second line of each table). The transport time appearing in column 4 is the time required for tangent slab transport. Relative time savings for the modi ed di erential approximation should be similar. Alternate values tested for these parameters are given in the rst column of the tables. The designations C1 and C2 refer to combinations of the alternate values as follows: C1 has na =75, nppl =11, nv =8, crit =0.0165 eV, and crit2 unchanged; C2 is the same as C1 except na =100. Three ight conditions identi ed as Cases A, B, and C, were studied to obtain results over a wide range of nonequilibrium conditions. These cases are described in further detail in Ch. 6 and 7. As shown in Tables 5.1-5.3, the prediction of the radiative ux is a ected by less than 5 percent for each set of parameters, except for two occurrences in Case C (see column 2). These are both attributable to setting na =75. The potential savings in computational time are considerable. Based on these results, it is determined that the parameter set denoted C2 (na =100, nppl =11, nv =8, crit =0.0165 eV, and
crit2 =0.0011 eV) is appropriate for use in the radiation model. Increasing crit2 in line with the changes in the other parameters was found to have no e ect, so results are not presented here.
5.2.2 Excitation Calculation The last column in Tables 5.1-5.3 shows the fraction of CPU time used by the radiation and transport calculations. The remainder is required to obtain the nonequi-
69
Table 5.1. Computational Optimization - FIRE II at 1631 sec Case
Flux
Baseline na =100 nppl =11 na =75 nv =8 crit 2 C1 C2
0.508 +0.8% +0.9% +0.8% +1.4% +0.8% +1.4% +1.4%
CPU time CPU time CPU for Radiation for Transport Fraction 6.116 1.735 0.50 3.561 0.895 0.37 -13.5% -29.7% 0.33 -1.6% -2.1% 0.36 -0.3% -0.8% 0.37 -0.9% -0.7% 0.37 -14.9% -30.8% 0.32 -26.6% -29.5% 0.30
Table 5.2. Computational Optimization - FIRE II at 1634 sec Case
Flux
Baseline na =100 nppl =11 na =75 nv =8 crit 2 C1 C2
22.50 +2.1% +3.6% +2.5% +2.6% +2.1% +4.5% +4.1%
CPU time CPU time CPU for Radiation for Transport Fraction 10.843 3.059 0.69 6.326 1.714 0.57 -13.0% -29.3% 0.52 -1.1% -1.6% 0.56 +0.1% -0.1% 0.57 -0.6% -0.8% 0.57 -14.8% -30.7% 0.52 -26.0% -29.3% 0.49
Table 5.3. Computational Optimization - FIRE II at 1637.5 sec Case
Flux
Baseline na =100 nppl =11 na =75 nv =8 crit 2 C1 C2
292.9 +1.5% +1.9% +6.1% +1.2% +1.6% +6.3% +1.7%
CPU time CPU time CPU for Radiation for Transport Fraction 10.061 2.913 0.73 5.754 1.617 0.60 -12.1% -28.9% 0.56 -1.2% -1.1% 0.60 -0.1% +0.1% 0.60 -0.7% +0.2% 0.60 -14.1% -30.2% 0.55 -24.5% -28.8% 0.53
70 librium excited state populations, and is a signi cant fraction of the total. The QSS algorithm used to perform this calculation was obtained from NEQAIR and is used as a black box. While important savings may be possible in this algorithm, no attempt has been made to achieve them. It is clear that this is a limiting factor in the eciency of the radiation prediction. It is interesting, but not surprising, to note that the excitation calculation takes longer for conditions which are further out of equilibrium.
5.2.3 Radiation Subgrid Calculating radiative properties at every point on a ow eld grid is often wasteful. The absorption and emission coecients only change signi cantly in regions where the temperatures or species concentrations have large gradients. Bolz [30] developed an algorithm to automatically select a subset of grid points for the radiation calculation. He de nes a weighting function which can be adapted to the nonequilibrium situation as
3 2 0 k 0 ji 0 ji 0 ji 0 ji X j q ~ j T j T N j T Zk = k+ 4WN 0 i + WTt jT 0j t + WTv jT 0 j v + WTe jT 0j e + WqR j~q0 jR 5 N i;max t i;max v i;max e i;max R i;max i=1 (5:1) The 0 in these equations denotes the partial derivative with respect to , the normal to the wall. The average derivative of the species number densities is
N 0i =
s X j =1
jNi;j0 j
(5:2)
71 12 500 LAURA grid 10000
Radiation subgrid
7500
T v, K 5000
2 500
0 0.0
2 .5
5.0
7.5
10.0
12 .5
15.0
η, cm
Figure 5.5. Example of Radiation Subgrid Selection for s chemical species. Radiation is then computed for kR < k grid points at equal intervals of Zk . An example of such a grid is given in Fig. 5.5. Though weighting factors are included for each term in Zk there is no obvious reason to emphasize the gradient in one variable more than that of another. These factors are therefore all set equal to one. This algorithm has been implemented in LORAN as an option. It can be invoked when reducing the computational time is important. If a separate rotational temperature is available, or if individual species vibrational temperatures are computed, additional terms can be added to the function Zk . The subgrid algorithm has been assessed using the three nonequilibrium test cases of Ch. 7. The e ects are summarized in Table 5.4. The subgrid results (columns 3 and 4) were obtained using the optimized C2 set of parameters discussed in Sec. 5.2,
72
Table 5.4. E ect of Radiation Subgrid Algorithm Case Detail spectrum 20 grid points 15 grid points All grid points A
0.508
-1.4%
+6.4%
B
22.50
+10.7%
+14.2%
C
292.9
+3.9%
+2.9%
while the detailed spectrum result (column 2) uses the Baseline parameter set to establish a reference heat ux (this is again a grid re nement test rather than an accuracy check, since even the detailed spectrum result relies on uncertain radiation and excitation properties). The CPU time required to compute the excitation and radiation for a given case is a nearly linear function of the number of grid points, so the time savings can easily be assessed. For each of the three cases using twenty grid points (out of 64 in the ow eld grid) selected with the modi ed Bolz algorithm provides radiative heat ux predictions within about ten percent of the reference result, even including the spectral approximations introduced by the use of the C2 data set. A subgrid of this size is therefore considered adequate, con rming Bolz' conclusions. In an axisymmetric or 3-D shock layer ow eld this algorithm can be applied to each normal ray of grid points. This results in a \patchwork" grid, in which cell shapes are more irregular than those in a standard LAURA grid. In an extreme case, such a grid may destabilize the transport solution. In most cases, however, optimizing
73
50
0
z, cm -50
-100 0
50
100
150
2 00
x, cm
Figure 5.6. Patchwork Grid Resulting from Application of Subgrid Algorithm Along Each Normal Line the subgrid along each normal ray may be acceptable and even bene cial. Figure 5.6 presents an example of such a grid for a test ow eld. The irregularities appearing in the mesh are in fact quite minor. To further reduce the required computational time, every other normal line of the
ow eld grid might be skipped in the radiation calculation. The required properties can easily be interpolated for the skipped normal lines.
5.3 Flow eld Coupling The divergence of the radiative ux, 5 ~qR, which appears in the energy conservation equations to couple radiation and uid dynamic phenomena, is computed for
74 each point in the ow eld by LORAN, either directly or by interpolation of results on a radiation subgrid. Coupling can be accomplished by alternatively computing the ow eld with LAURA and the radiative ux term with LORAN. Because the computation of the radiation requires up to three orders of magnitude more computational time than one iteration of the uid dynamics computation, however, it is advantageous to utilize a multitasking strategy. Multiple tasks are assigned to the radiation calculation while a single task computes the nonequilibrium ow eld. The computational domain is partitioned into three to seven subdomains (depending on the number of processors available), each assigned to a processor for calculation of the radiation eld. When run asynchronously with the single ow eld task, the multiple radiation tasks allow the ow eld equations access to the latest radiative ux terms in shared memory (multiple LAURA iterations still occur during a single radiation update). This asynchronous strategy is an ecient use of a multiple processor machine since no processor remains idle waiting for another task. Preliminary computational experiments indicate no instabilities in this approach. The e ectiveness of asynchronous multitasking schemes using LAURA is further discussed in Ref. [75]. As a practical matter, it is often more ecient to rst obtain a converged ow eld solution using LAURA, then \turn on" radiation and converge the coupled ow eld. If the radiation is only a small perturbation to the ow eld this second convergence may require only a few additional LAURA iterations. If there are strong radiation e ects then converging the perturbed ow eld will require many additional iterations.
75 Though the necessary code to couple the LAURA and LORAN solutions has been generated, only a very few preliminary results have so far been obtained with
ow eld coupling. The cases run to date con rm that this coupling can be done, but no signi cant e ects have yet been observed for the cases so far examined. Coupled results are therefore not included in Ch. 6.
76
6 Results and Discussion Gas radiation is a complex phenomenon, so that true veri cation of a radiation model requires checking it against a large number of ight and ground-based experiments covering a wide range of conditions. Only a few such measurements are currently available. Representative results for these cases are shown here as an initial test of the method. All nonequilibrium ow elds used in the veri cation were generated by the LAURA code of Gno o [71], in order to eliminate di erences caused by
ow eld modeling. As discussed in the following chapter, the selection of models in a ow eld code can have a large impact on the predicted radiation.
6.1 Comparison to Experiment 6.1.1 Ground-Based Data: AVCO Shock Tube One of the classic radiation measurements is that in the AVCO shock tube [43]. Using the gas conditions suggested by Park [72], a predicted spectrum for the emission intensity has been obtained for the peak radiation point, and is shown in Fig. 6.1. This spectrum can be compared to Figure 5 of the just-cited reference, which is reproduced here in Fig. 6.2 and shows the measured spectrum compared to a prediction obtained by NEQAIR. The agreement between the measured data and the present result is no worse than that shown by NEQAIR and may actually be a little better. The atomic lines appearing in Fig. 6.1 are multiplet-averaged, and so should exhibit only a gross
77 agreement with the NEQAIR prediction of Fig. 6.2, as is observed. It is clear from Fig. 6.2 that the measurements obtained in the AVCO experiments had a lot of scatter. It is dicult, therefore, to do more than note the general agreement between the experimental results and the prediction. Both predictions do have signi cant di erences from the AVCO data in a few spectral regions (particularly below 0.3 m). These may be attributable either to contamination of the shock tube or to inexact speci cation of the gas conditions.
6.1.2 Flight Data: Project FIRE The FIRE ight project of the mid-1960's consisted of two ights simulating reentry of an Apollo type vehicle. The rst ight, FIRE I, experienced telemetry problems that made data reduction and analysis dicult as well as control problems during the second half of the entry that resulted in substandard data. These problems were corrected before the FIRE II ight, which provided good measurements of the total and spectral radiation at the stagnation point during the forty second entry period [76, 77]. The FIRE ight vehicles had an Apollo-like geometry with a layered beryllium heatshield (the second layer had a geometry identical to Apollo). Each of the three layers was used up to a temperature limit and jettisoned, resulting in three periods of prime data during the entry. The rst heatshield had a nose radius of 0.935 m and a diameter of 0.672 m. Only data obtained during the use of this rst heatshield will be examined here, since it coincided with the nonequilibrium portion of the ight.
Emission Intensity, W/cm3 -µm-sr
78 10
2
10
1
10 0
10 -1
10 -2
10
-3
0.2
λ, µm
10
0
2.
Figure 6.1. Calculated Spectrum for the Peak Radiation Point of the AVCO Experiment
Figure 6.2. Measured Spectrum and NEQAIR Prediction for the Peak Radiation Point of the AVCO Experiment
79 The trajectory of the FIRE II ight is reported in Lewis and Scallion [62]. Forebody temperature histories are given in Cornette [78]. The instrumentation used for the ight measurements is described by Richardson [61]. In particular, the spectral response of the radiometer windows is reported by Dingeldein [79, Fig.5]. It is at between 0.23 and 2. m in wavelength, falling o sharply below 0.2 m and more gradually above 2. m. The design goal is reported as a at response from 0.2 to 6. m. The various reports on FIRE quote actual spectral ranges starting between 0.2 and .23 m, and ending between 4. and 6. m. Taking these variations into account, the spectral range of 0.23 to 4. m (0.31 to 5.4 eV) is considered to be closest to the actual window transmission range. FIRE also had scanning spectral radiometers that were designed to cover the range of 0.2 to 0.6 m (2.1 to 4.1 eV). Mechanical problems during the ight limited forward scans to 0.3-0.5575 m, and backward scans to 0.6090-0.3 m [77]. The spectral resolution is quoted as 0.004 m, with a root-sum-square uncertainty in the measured spectra of 23 percent. This resolution is insucient to resolve any atomic lines. For this study of stagnation line radiation, the FIRE vehicle geometry has been modeled by a sphere with an e ective nose radius of 0.747 m. A fully catalytic wall boundary condition at the ight-measured wall temperature was used. The cases selected from the FIRE II ight are shown in Table 6.1 [62]. The designations Case A, B and C will be used below to identify each case. These cases cover the range from extreme nonequilibrium to near-equilibrium according to Cauchon [76,
80
Table 6.1. Selected FIRE II Flight Conditions Case Time Altitude Velocity Density
Tw
(km/sec) (kg/m3) (K)
qR
(sec)
(km)
(W/cm2)
A
1631
84.59
11.37
9.15e-6
460
.1620%
B
1634
76.41
11.36
3.72e-5
615
8.220%
C
1637.5
67.04
11.25
1.47e-4 1030 81.720%
p.14]. Predicted temperature pro les along the stagnation line, using the baseline set of energy exchange models in LAURA (de ned in Ch. 7), are presented in Figs. 6.36.5. For Case A, extreme thermal nonequilibrium is predicted, in agreement with Cauchon, while Case C is predicted to be in thermal equilibrium through about half the shock layer. The ight radiation levels (qR) quoted in the above table were taken from Cauchon [76, g.13] and converted from intensities to uxes assuming a transparent planeparallel shock to be consistent with the calculations below. Also shown is the rootsum-square uncertainty estimated in a post- ight error analysis. It should be noted that the uncertainty in the radiation level for Case A is probably higher, as the radiometers were at the lower limit of their sensitivity range for that case. The measurements show considerable scatter during the early part of the trajectory [76], with a variation of about a factor of three between high and low readings for the rst several seconds. The scatter decreases as the level of radiation increases. It is noteworthy that the FIRE I data acquired in this density range for a slightly higher
81
50000
40000
Tt Tv
30000
T, K 2 0000
10000
0 0.00
0.04
y, m
0.08
0.12
Figure 6.3. Predicted FIRE II Stagnation Line Temperature Pro les - 1631 sec 60000
Tt Tv 40000
T, K 2 0000
0 0.00
0.04
η, m
0.08
Figure 6.4. Predicted FIRE II Stagnation Line Temperature Pro les - 1634 sec
82 40000
Tt Tv
30000
T, K 2 0000
10000
0 0.00
0.02
0.04
0.06
y, m
Figure 6.5. Predicted FIRE II Stagnation Line Temperature Pro les - 1637.5 sec velocity (11.57 vs 11.37 km/sec) were higher by a factor of three than the FIRE II results for both the total and spectral radiometers. The two ights thus suggest a very sensitive dependence of radiative heating on freestream velocity under these ow conditions. At the conditions of Case B, the ight-to- ight variation is about a factor of 1.6. For Case C, it is about 1.3. Detailed comparisons between the LORAN predictions and the FIRE ight radiometer measurements are given in the following chapter, as part of a study in which the impact of various nonequilibrium ow eld models on radiation is assessed. Some additional comparisons between LORAN and other nonequilibrium radiation prediction methods are given here. The predicted pro les of radiative emission along the stagnation line for Case B, for
83 10.0 atomic cont. atomic line mol. band total atomic cont. atomic line
E, W/cm3
7.5
mol. band total
5.0
}
LORAN
}
NEQAIR
2 .5
0.0 0
1
2
η, cm
3
4
5
Figure 6.6. Emission Pro les for FIRE II at 1634 sec with Comparison to NEQAIR the spectral range in which the radiometer windows transmit, are shown in Fig. 6.6. Calculations from LORAN and NEQAIR [20] are compared, broken out into the various radiating mechanisms. The NEQAIR result shown here was computed using a version of the NEQAIR code which was obtained from Chul Park in 1988. It has been slightly modi ed to correct some minor errors this version contained, and to allow emission calculations within a limited spectral range. The agreement between the two predictions is excellent except for a discrepancy in the atomic line radiation near the wall. The predicted wall radiative heat ux di ers by only 3 percent between the two codes. Figure 6.7 shows the emission pro les for Case C. The overall agreement between
84 12 5 atomic cont. atomic line molecular band
100
}
LORAN
}
NEQAIR
E, W/cm3
total atomic cont. atomic line
75
molecular band total
50
25
0 0
1
2
η, cm
3
4
5
Figure 6.7. Emission Pro les for FIRE II at 1637.5 sec with Comparison to NEQAIR the two predictions is quite good, although there are some di erences. The radiative
ux reaching the wall varies by about 6 percent between these two predictions. Radiation predictions for these cases have also been made by Carlson and Gally [35] for a nitrogen freestream. Figure 6.8 can be qualitatively compared to Fig. 6.9, which is reprinted from Figure 5 of their paper. It shows the approximate spectral variation of the wall radiative ux over the entire spectrum including the ultraviolet. Though the predicted magnitudes are lower in Ref. [35] because only nitrogen is considered, the variations are nonetheless qualitatively very similar. The details of the spectra cannot be compared because of the line group presentation used in the reference. It should also be noted that the chemical kinetics and energy exchange
85
Table 6.2. Radiative Heat Flux Predictions for FIRE II Case
NEQAIR
LORAN
Flux, W/cm2 (0.31-16.5 eV)
Di erence
LORAN
Percent
with Absorption
1634 sec
1682.
1806.
7.4
29.35
1637.5 sec
13233.
14771.
11.6
421.2
models used by Carlson and Gally are di erent from those in the LAURA code. As demonstrated in Ch. 7, these models can have a large impact on the magnitude and spectral distribution of the predicted radiative heating. Results have also been computed for the complete radiation spectrum (0.31 to 16.5 eV) for these two cases. These are summarized in Table 6.2, which compares the emission predictions between NEQAIR and LORAN. The di erence, about ten percent, is minor. The last column shows the total radiative ux reaching the wall when absorption is included in the transport calculation. The large self-absorption of the ultraviolet portion of the spectrum is evident here. This wall ux is still somewhat high relative to the ight data, but the e ect of radiation cooling on the ow eld has not been accounted for in these calculations.
6.2 Nonequilibrium Test Cases 6.2.1 Mars Return A ow eld solution has been obtained for one of a number of possible ight conditions identi ed for the return from a mission to Mars [80]. It consists of a 60o sphere
86
12 5 1634 sec 1637.5 sec
2
qR , W/cm -eV
100
75
50
25
0 0
5
10
15
hν, eV
Figure 6.8. Spectral Variation of Stagnation Point Radiative Heat Transfer for FIRE II
Figure 6.9. Spectral Variation of Stagnation Point Nitrogen Radiation for FIRE II
87 cone with a 1.08 m nose radius ying at 80 km altitude with a velocity of 12 km/sec. The radiative heating for this case is not too severe, so it provides a good initial test of the method. Results have been obtained for this case using both tangent slab transport and the modi ed di erential approximation (MDA). Figure 6.10 compares the wall radiation ux predictions from the two transport methods. As expected, the MDA result is lower than the tangent slab result. The low qw value for the MDA result near the stagnation point is believed to be erroneous, however. The sudden increase in qw at the shoulder in the MDA result certainly is, and is attributable to the treatment of the out ow boundary. Further work will be required to correct both of these problems. The qualitative trends of the two methods are similar, showing the radiative ux increasing with distance from the stagnation point. This increase is unlike the behavior of convective heating on such a body and results because radiation is a volumetric quantity. As seen in Fig. 6.12 below, the stando distance is much larger on the ank than at the stagnation point, and the region of signi cantly radiating gas has increased greatly. The vibrational temperature in this nonequilibrium ow eld is still high in this region, and a large amount of radiation is therefore emitted from the gas. The other variable of interest for coupled ow elds is the divergence of the radiative ux in the shock layer. Figures 6.11 and 6.12 show contour plots of 5 ~qR computed with each transport method. The tangent slab result shown in Fig. 6.11 is in fact just the derivative of qR along each normal grid line, so the values at neigh-
88 30 Tangent slab
qw, W/cm
2
25
MDA
20 15 10 5 0 0
50
100
150
2 00
2 50
s, cm
Figure 6.10. Wall Radiative Flux for Mars Return Case
boring grid lines do not in uence each other. Compared to the MDA result given in Fig. 6.12 the tangent slab result is very disjointed. Accounting for the long range in uence of radiation with the MDA solution provides a much smoother variation of
5 ~qR. Should stability be an issue in a coupled ow eld solution, therefore, the MDA result might be expected to have less of a destabilizing e ect than the tangent slab result. While this has not traditionally been used as an argument against the tangent slab approximation, it may prove to be decisive at some ow conditions. For the few coupled cases run to date, the stability of the CFD solution has not been reduced by adding radiation.
89
dqR /dn
50
3
(W/cm ) 70.00 50.00 30.00 10.00 -10.0
0 z, cm
-50
-100 0
50
100
150
2 00
x, cm
Figure 6.11. Radiative Flux Divergence for Mars Return Case with Tangent Slab Transport ∇⋅qR
50
3
(W/cm ) 70.00 50.00 30.00 10.00 -10.0
0 z, cm
-50
-100 0
50
100
150
2 00
x, cm
Figure 6.12. Radiative Flux Divergence for Mars Return Case with MDA Transport
90
6.2.2 Aeroassist Flight Experiment The Aeroassist Flight Experiment (AFE) is a NASA project intended to obtain
ight data for the design of future Aeroassist Space Transfer Vehicles (ASTV). Measurement of the nonequilibrium radiative heating to the AFE surface is one of the principle objectives of the project. The aerobrake con guration is a raked cone with a blunted elliptic nose [81]. Several studies of nonequilibrium radiative heating for this con guration have been made to provide inputs to the design of the heatshield [17, 37, 82, 83]. The LORAN method has been used to obtain a prediction of the level and distribution of radiative heating at the peak heating point in the trajectory. An axisymmetric LAURA ow eld solution in which the AFE geometry was modeled by a sphere with an equivalent nose radius of 2.16 m [84] was generated [85]. The ow eld models used to obtain this solution are the baseline models listed in Sec. 7.1, including the use of the Park-87 chemical kinetics. The wall temperature was assumed to be constant at 1750 K. Radiation predictions were obtained for this case, again using both tangent slab and MDA transport methods. The tangent slab calculation was completed in about 45 minutes of actual elapsed time on a Silicon Graphics 4D/320, while the MDA result required about 12 hours of elapsed time. The convergence of the MDA solution for this case was much slower than that for the Mars return case (in fact this case is not quite converged), but even this run time is not unreasonable. The radiative heating rates
91 5.0
qw, W/cm2
4.0
3.0
2.0
Tangent slab MDA
1.0
0.0
0
10
20
30
40
50
s, cm
Figure 6.13. Wall Radiative Flux for AFE predicted for the AFE peak heating point are given in Fig. 6.13. These predictions are in the same range as earlier computations for the AFE radiative heating [4, 86]. The MDA prediction is again lower than the tangent slab result, with problems still appearing at the grid boundaries. It is expected that running the MDA transport to complete convergence for this case would result in closer agreement between the two, as would a correction for the depressed MDA value near the stagnation line. It should also be noted that only the nose region of the equivalent sphere ow eld has been used for the AFE case. Away from the nose the geometry deviates quickly from the AFE con guration so the results are not of interest. The radiative ux divergence for the AFE ow eld is presented in Figs. 6.14 and 6.15 for the tangent slab and MDA transport, respectively. Again the smoother
92 behavior of the MDA solution can be noted. As mentioned earlier, the tangent slab method employs a numerical di erentiation to obtain dqR=dn. This di erentiation may be adding to the error already incurred by ignoring the variation of the radiation properties in the direction parallel to the surface. The amount of heating for the AFE vehicle resulting from radiation in the ultraviolet (UV) portion of the spectrum has been the subject of some debate. The distribution of radiation predicted by the LORAN method is shown in Fig. 6.16. Curves are shown using the tangent slab and MDA transport models. The relative importance of di erent spectral regions varies only slightly between the two transport methods. Both curves indicate that just over half of the radiative heating experienced by the AFE spacecraft results from the UV portion of the spectrum. This is a significant fraction of the total. Radiation in the UV spectral range is highly self-absorbed (see for example Table 6.2). If the amount of self-absorption is mispredicted only slightly, the wall radiative ux may be signi cantly increased for AFE.
93
20
dqR /dn 3
(W/cm ) 15.00 10.00 5.00 0.00 -5.00
10
z, cm
0
0
5
10
15
20
25
30
35
x, cm
Figure 6.14. Radiative Flux Divergence for AFE with Tangent Slab Transport 20
∇⋅qR 3
(W/cm ) 15
15.00 10.00 5.00 0.00 -5.00
10
z, cm 5 0 -5 0
5
10
15
20
25
30
35
x, cm
Figure 6.15. Radiative Flux Divergence for AFE with MDA Transport
94
5.0
Tangent slab
ν
∫0 qν dν
4.0
MDA
3.0 2.0 1.0 0.0
0
5
10
15
20
hν, eV
Figure 6.16. Spectral Distribution of AFE Wall Radiative Flux
95
7 Flow eld Model Studies Flow eld studies were performed using the Langley Aerothermodynamic Upwind Relaxation Algorithm (LAURA) code developed by Gno o [71] to investigate the sensitivity of the radiation predicted by LORAN. LAURA, like all nonequilibrium Computational Fluid Dynamics (CFD) codes, contains a number of semi-empirical models governing the exchange of energy between energy modes. LAURA presently incorporates the two-temperature model of Park, in which the heavy particle translational and rotational energy modes are assumed to equilibrate to a temperature Tt (the combination is referred to as the translational mode), and the vibrational, electronic, and free electron translational energy modes are assumed to reach a separate equilibrium with a temperature Tv (the combination is referred to as the vibrational mode). Since radiation is strongly in uenced by the amount of energy available in each mode, a study was conducted to assess the impact that uncertainties in these energy exchange models have on radiation predictions. The ow eld cases chosen for these studies are taken from the trajectory of the FIRE II ight project [61, 62]. Widely varying freestream conditions were selected to exercise the exchange models over a range of levels of nonequilibrium. A description of this ight experiment was given in Sec. 6.1.2. All the cases run for the ow eld studies had 64 grid cells normal to the body in the LAURA solution. This is not sucient to resolve the shock, as illustrated for instance in Fig. 6.5. One grid-resolved solution
96 with 128 grid cells was obtained to quantify the e ect of this lack of resolution. The radiation prediction obtained with this re ned grid was only about 15 percent di erent from that with the coarser grid. This is not enough to change any of the conclusions drawn from this study. All the results presented in this chapter are for the stagnation line of the FIRE II vehicle and use the tangent slab radiative transport method. Only the spectral range in which the FIRE radiometer windows were transparent (0.31-5.4 eV) has been included.
7.1 Baseline Models A discussion of the energy exchange mechanisms and the alternate models used for each is given in the following sections. The set of models considered as the baseline consists of the most recent recommendations from Park which have been incorporated in the LAURA code. They are, for the vibrational-translational energy exchange cross section:
v = 10?21 (50; 000=Tt )2 m2
(7:1)
for the dissociation temperature:
Td = Tt:7 Tv:3
(7:2)
and for the energy exchange in dissociation: Ev = Ev
(7:3)
97 In addition, the set of chemical reactions known as Park-87 [87] is considered part of the baseline. A run identi er speci es the models used for each solution. The baseline set is denoted v173. The rst two characters, v1, denote the rst model for vibrationaltranslational energy exchange, Eq. 7.1. The last two, 73, denote the powers in the dissociation temperature equation. Only alternate models for the energy exchange in dissociation and the chemical reaction set are agged in the run identi er for conciseness. These will be explained as they are used below.
7.2 Energy Relaxation The modeling of relaxation or equilibration between energy modes is an important question in nonequilibrium CFD. Generally the process of equilibration is not completely understood and must be described semi-empirically. Several formulations have been proposed for many of these models. Data to evaluate the di erent formulations are scarce. To quantify the in uence which the di erent proposed formulations of the models for energy relaxation have on predictions of radiative heat transfer, a parametric study was undertaken on a number of such models. The models and formulations considered, and the results obtained, are detailed below.
98
7.2.1 Dissociation Temperature The dissociation temperature, Td, is the rate-controlling temperature for reactions involving the dissociation of a molecular species. Several formulations have been proposed for this temperature, notably by Park and his co-workers [87, 88]. These models are empirical and consist of a geometric weighting of the translational temperature
Tt and the vibrational temperature Tv as follows: Td = TtmTvn
(7:4)
The choice of powers m and n adjusts the weight given to each temperature. The recent trend has been to increase the weighting of the translational temperature Tt because of indications that heavy particle collisions are more important than other mechanisms in causing dissociation. Three di erent models with (m; n) equal to (.5,.5), (.7,.3), and (1,0) are considered here. The rst set is the original Park twotemperature proposal, the second the more recent model, and the third is included to study the extreme case of dissociation controlled by the heavy particle temperature alone. The dissociation temperature models will be denoted by 55, 73, and 10, respectively, in the third and fourth digits of the run identi er. As shown in Figs. 6.3-6.5, the translational temperature Tt is much higher than
Tv near the shock. Increasing the weighting of the dissociation temperature, Td, on Tt in Eq. 7.4 therefore results in faster dissociation of the molecular species. This dissociation removes both translational and vibrational energy, decreasing Tt and Tv . The increased dissociation is shown in Fig. 7.1 which compares the N2 and O2 pro les
99 along the stagnation line for Case A for the two extreme Td models. The e ect on the temperature pro les is shown in Fig. 7.2 also for Case A. The decreased temperature results in increased density and a smaller shock stando distance. For the three FIRE II cases, placing increasing weight on the translational temperature Tt in the de nition of the dissociation temperature Td decreases the total radiation. Part of the decrease occurs in the molecular bands, which depend directly on the molecular concentrations. The rest of the decrease results from the lower Tv and decreased stando distance a ecting all three radiating mechanisms. Figure 7.3 shows the emission pro les for the three Td models for Case A. Case A is the most nonequilibrium of the three FIRE conditions studied here, and therefore is most sensitive to these energy relaxation models. The sensitivity to the Td model decreases at the later, more equilibrium, ight times of Cases B and C. Tables 7.1-7.3 at the end of this chapter reveal that in all three cases (v110, v155 and v173) the predictions are closer to the ight data when Tt is weighted more in the de nition of Td, con rming the recent work on this model mentioned above. For Cases A and B, agreement within about a factor of three is found. For Case C, the case nearest equilibrium, the variations caused by this model are within the uncertainty of the ight data.
7.2.2 Vibrational-Translational Energy Exchange The equilibration of the vibrational and translational energy modes is modeled using a relaxation time, v , for each species. Millikan and White [89] proposed a
100 10
17
10 16
N, cm-3
10 15 10
14
10
13
N2 O2
10 12 10
11
10
10
0.00
N2 O2
0.04
y, m
0.08
}
v110
}
v155
0.12
Figure 7.1. E ect of Td Models on Molecular Dissociation - 1631 sec semi-empirical formulation, vMW , in the range of 300 to 8000 K. Park [90] suggests an additional collision limiting correction, vP , for high temperatures where the MillikanWhite correlation predicts excessively fast relaxation. A concise explanation of the failure of the Millikan-White correlation is provided by Sharma and Park [91, p. 136]. The total relaxation time is then
v = vMW + vP
(7:5)
vP = (v vsNs)?1
(7:6)
where Park's contribution is In this expression vs is the average speed and Ns the number density of molecular species s, and v is an e ective cross section for vibrational relaxation. The latter quantity has been the subject of some debate, with three di erent values receiving
101
60000 v110 v155 40000
v173 Tt
T, K 2 0000 Tv 0 0.00
0.04
y, m
0.08
0.12
Figure 7.2. E ect of Td Models on Temperature Pro les - 1631 sec 1.5
E, W/cm3
v110 v155 v173
1.0
0.5
0.0 0.00
0.04
0.08
0.12
y, m
Figure 7.3. E ect of Td Models on Radiative Emission Pro les - 1631 sec
102 support in various e orts to match the limited experimental data that has bearing on the question:
v = 10?21 (50; 000=Tt )2 m2
(7:7)
v = 10?20 m2
(7:8)
v = 10?21 m2
(7:9)
Equation 7.8 is the original proposal, which assumes this cross section to be one tenth of the elastic cross section. Equation 7.9 was suggested when Eq. 7.8 seemed still too high. Equation 7.7 is a modi cation of Eq. 7.9 that reduces the contribution of vP at low temperatures. All three models are considered here, and are identi ed by the notation v1, v2, and v3, respectively, in the rst two digits of the run identi er. A further modi cation to this energy exchange process has been proposed by Park [87]. He argues that vibrational relaxation exhibits a di usion-like behavior at high temperatures which requires a correction to the relaxation time. However, this modi cation requires evaluation of the post-shock levels of Tt and Tv . Interpretation of these quantities in a shock capturing solution is somewhat ambiguous [92]. Improper de nition can lead to instabilities in some circumstances, consequently no attempt was made to include this modi cation. Figures 7.4 and 7.5 present the e ect of the di erent vibrational-translational energy exchange cross sections in Eqs. 7.7-7.9 on the temperature and radiative emission pro les predicted for Case A. For a smaller v , the relaxation time v increases. Increasing v delays the relaxation of translational to vibrational energy. This increases
103 60000 v173 v2 73 40000
v373 Tt
T, K 2 0000 Tv 0 0.00
0.04
0.08
0.12
y, m
Figure 7.4. E ect of v Models on Temperature Pro les - 1631 sec Tt while reducing Tv , as illustrated in Fig. 7.4. Although the stando distance increases with increasing Tt, the reduction in radiation due to the lower Tv in this model more than compensates. The lower Tv reduces the radiation from all three mechanisms, as shown in Figure 7.5. The lowest radiation for all three FIRE cases is predicted with Eq. 7.9, and the highest with Eq. 7.8. Comparing the predicted radiation for the v173, v273, and v373 models in Tables 7.1-7.3 shows that the in uence of v is largest in Case A, the most nonequilibrium case. Again all three cases are closest to FIRE II when the radiation prediction is lowest, with Eq. 7.9, though only Case A exhibits much sensitivity or discrimination among the models. The predictions in Case B remain high, while the e ect on Case C, the near-equilibrium case, is within the data uncertainty as was observed for the
104 2 .0 v173 v2 73 v373
E, W/cm3
1.5
1.0
0.5
0.0 0.00
0.04
0.08
0.12
y, m
Figure 7.5. E ect of v Models on Radiative Emission Pro les - 1631 sec dissociation temperature models.
7.2.3 Energy Exchange in Dissociation When a diatomic molecule dissociates, the vibrational energy it contained is consumed by the higher ground states (heats of formation) of the constitutive atoms. The energy thus removed must be accounted for in the vibrational energy equation. The amount of energy lost is commonly assumed to be the average vibrational energy at the local conditions, Ev . Because dissociation from a higher vibrational state may be more probable (the concept of preferential dissociation) several other formulations have been proposed [69]. One model assumes that the vibrational energy loss in dissociation, Ev , is some
105 fraction of the dissociation energy of a molecule measured from its ground state, D: Ev = c1D
(0 < c1 1)
(7:10)
Sharma et al. [88] have proposed using c1 = 0:3. This model was considered, but since it predicts the non-physical result of a negative Tv in the shock region it was not pursued [92]. Park [93] has proposed that Ev = D ? kTt
(7:11)
so that the energy lost is the dissociation energy minus the average translational energy. This model arises from the assumption that dissociation only occurs from levels that are within the average collisional energy of the dissociation threshold (similar to the mechanism of bound-free radiation). Again, this Ev is too large (particularly when Tt is low) and results in an unrealistically small Tv behind the shock [92]. A third proposed model assumes that dissociation occurs from some vibrational state(s) above the average. This is expressed as: Ev = c2Ev
(c2 > 1:)
(7:12)
The best value of c2 to use in this empirical model is unknown, and may depend on
Ev (or Tv ). In this work c2 = 2 has been selected to provide an initial assessment of the model. The sux 2evs is used in the run identi er for these cases.
106 50000
40000
v173 2 evs Tt
30000
T, K 2 0000
10000
0 0.00
Tv
0.02
0.04
0.06
0.08
y, m
Figure 7.6. E ect of Ev Models on Temperature Pro les - 1634 sec Figure 7.6 compares the temperature pro les for Eq. 7.12 with c2 = 1 (the baseline model) and c2 = 2 for Case B. The increased Ev in the 2evs model (c2 = 2) reduces
Tv in the region behind the shock where dissociation occurs, and increases it deeper in the shock layer where recombination begins. It also results in a smaller shock stando distance. Figure 7.7 presents the radiative emission for Case B and con rms that the lower Tv near the shock results in reduced radiation, while the higher Tv in the shock layer increases it. This is observed in the tables as a decrease in the molecular band radiation and a slight increase in the atomic contributions, resulting in a net decrease in the total radiation. (The lower peak in the dashed curve of Fig. 7.7 is the atomic radiation peak.) For Cases A and B, the decreased radiation resulting from the 2evs model brings
107 18 v173
15
E, W/cm3
2 evs 12 9 6 3 0 0.00
0.02
0.04
0.06
0.08
y, m
Figure 7.7. E ect of Ev Models on Radiative Emission Pro les - 1634 sec the prediction closer to the ight measurement, but it remains high (Tables 7.1-7.3). For Case C the e ect is within the data uncertainty, but this is the only model where the prediction is lower than the ight measurement. Table 7.3 also includes a solution run with the model of Eq. 7.10 using c1 = 0:3 (denoted 3dis) for comparison. A ag was introduced in LAURA to suppress negative temperatures. The result is not signi cantly di erent so it provides no incentive to pursue this model. As with the two previous energy exchange models, the sensitivity to this model decreases as the ow eld tends toward equilibrium. The decreasing sensitivity to all these models near equilibrium is expected, since they govern the exchange between energy modes in the two-temperature thermal nonequilibrium model. When dealing with ow elds near equilibrium, the choice of formulation for these models is not as
108 important. The remaining models shown in Tables 7.1-7.3 will be discussed later.
7.3 Spectral Radiation Comparison To obtain further information on the validity of the energy exchange models, the FIRE spectral radiometer measurements (vs. wavelength, , in m) are compared with those exchange models whose predictions are closest to the ight measurements. Only continuum radiation is shown for the predicted spectra, since the resolution of the ight spectral radiometer is not sucient to record any line radiation. In evaluating the ight spectra reproduced here, it should be noted that only a few are available in the literature. Those that have been published are the ones that were judged by researchers at the time to contain the least amount of noise. This criterion had the e ect of selecting spectra that correspond to the lower range of ight radiation measurements. It is not known whether the unpublished data which correspond to the higher range of measurements had increased intensity levels or broader spectra or both.
Case A (t=1631 sec) The spectral distribution of radiation has been examined for Case A for the v373 solution, which appears in Table 7.1 to best match the ight data. Figures 7.8 and 7.9 compare the prediction at 1631 sec with a ight spectral radiometer scan at 1631.3 sec, the closest available, reproduced from Cauchon [77]. Though the prediction is much higher than the ight scan they are qualitatively similar. Both show radiation from
109 the N2+ rst negative band, whose two major band heads can be distinguished. (The spike in the ight spectrum between 0.5 and 0.6 m has been attributed to a spurious signal [77].) Integrating the LORAN predictions over this limited spectral range results in a value that is high but within the data scatter at this ight condition [76, Fig. 13].
Case B (t=1634 sec) Figures 7.10 and 7.11 compare the 2evs result for Case B, the closest to the ight data in Table 7.2, to a ight spectrum at 1634.43 sec [77], the closest time available. The qualitative agreement is comparable to that shown for Case A above. Integrating the predicted radiation over the range of the spectral radiometer gives a result that again, while high, is within the scatter of the ight data given in Fig. 13 of Cauchon [76]. The curve labeled dk73 in this gure will be discussed in Sec. 7.4.
Case C (t=1637.5 sec) Figures 7.12 and 7.13 compare the predicted v373 spectrum for Case C to the nearest available ight spectrum, at 1636.43 sec. As in Case A, the v373 shows the closest match to the ight data in Table 7.3. By this point in the trajectory, the radiometer windows are approaching their melting point. A detailed post- ight analysis determined, however, that the windows cause less than a 10 percent change in the measured radiation [76]. The intensity spectrum predicted by the v373 is excessive, having far too much radiative energy in the N2+ rst negative band. The
110
10
2
I λ, W/cm2 -ster-micron
v373
10
1
10
0
-1
10 0.2 0
0.30
0.40
λ, microns
0.50
0.60
Figure 7.8. Predicted Radiation Spectrum - 1631 sec
Figure 7.9. Measured Radiation Spectrum - 1631.3 sec
111
10
2
I λ, W/cm2 -ster-micron
2 evs dk73 10 1
10
0
-1
10 0.2 0
0.30
0.40
λ, microns
0.50
0.60
Figure 7.10. Predicted Radiation Spectra - 1634 sec
Figure 7.11. Measured Radiation Spectrum - 1634.43 sec
112 curve labeled v1eq will be discussed below.
7.4 Nonequilibrium Chemistry The set of chemical reaction rates used in a ow eld model is somewhat better de ned than are the rates of energy relaxation. Uncertainties remain, however, particularly in the extrapolation of the rates to temperatures at which little or no experimental data is available. Two rate sets are in widespread use: that of Dunn & Kang [94], and that of Park [87]. The latter rate set has been updated several times [67, 95]. In addition, there are several options for obtaining rates for the backward reactions. Dunn & Kang originally included curve ts for the backward rates. It is generally agreed, however, that more accurate results for backward rates are obtained by computing them from the forward rates and associated equilibrium constants (at appropriate temperatures). These equilibrium constants are commonly obtained from curve ts. Recently Gupta [96] generated a new set of curve ts for these constants which di er noticeably from those proposed by Park. Mitcheltree [97] has used the LORAN radiation model in the LAURA code to study the in uence of these various sets of chemical rate data for a 12 km/sec entry. A few results for the FIRE II cases are presented here. The Dunn & Kang chemical kinetics model, denoted dk, has been applied to each of the FIRE II cases. As shown in Table 7.2, it results in a closer match of the ight heating rate for Case B. The wall spectral intensity predicted for this model was therefore also included in Fig. 7.10. This spectrum is much closer to the FIRE II
113
I λ, W/cm2 -ster-micron
10
2
v373 v1eq
10 1
10
0
-1
10 0.2 0
0.30
0.40
λ, microns
0.50
0.60
Figure 7.12. Predicted Radiation Spectra - 1637.5 sec
Figure 7.13. Measured Radiation Spectrum - 1636.43 sec
114 50000
v173
40000
dk73 30000
Tt
T, K 2 0000
Tv
10000
0 0.00
0.02
0.04
0.06
0.08
y, m
Figure 7.14. E ect of Chemical Kinetics Model on Temperature Pro les - 1634 sec measurements than any of those obtained for Case B with the Park kinetics [87]. The distribution and relative magnitudes of the several band heads in this gure agree quite well with the ight data, and the Dunn & Kang prediction exhibits quantitative as well as qualitative agreement with the measured spectrum. For Case A, Table 7.1 shows that the Dunn & Kang model is not an improvement. For the near-equilibrium Case C, the Dunn & Kang result was obtained for a thermal equilibrium solution. This resulted in a signi cantly higher radiation prediction (Table 7.3). Referring again to the temperature pro les in Figs. 6.3-6.5, it is clear that the thermal relaxation is in two stages with di erent time constants. Right behind the shock Tt decreases rapidly as a result of dissociation, then starts to level o . This suggests the involvement of slow reactions such as ionization, which become increasingly
115 important as the level of ionization rises. This two-stage equilibration is much less apparent for the Dunn & Kang chemical kinetics model pro les shown in Fig. 7.14 for Case B. The improved agreement noted for Case B only with the Dunn & Kang solution suggests, therefore, that this second equilibration process deserves further study.
7.5 Nonequilibrium Temperature The most obvious model of importance to the radiation calculation is the nonequilibrium temperature model. The choice of the number of temperatures modeled and the equations used is a crucial one for radiation. In a completely nonequilibrium situation, separate temperatures would be required for each energy mode of each species. This is impractical, except perhaps in a Direct Simulation Monte Carlo (DSMC) solution. Park's two-temperature model, discussed above, seems to provide reasonable accuracy with the minimum of complexity. If three temperatures are to be used, there is as yet no consensus about the electron energy equation which should be employed. Carlson and Gally [98] have studied this question for Martian return conditions. Candler [4] has studied the problem for general conditions. For Case C, which is close to thermal equilibrium, solutions were generated for both the Park and the Dunn & Kang chemical kinetics, using Park's two temperature model or assuming thermal equilibrium (a single temperature T ). This allows an assessment of the extent and impact of thermal nonequilibrium. The results are included in Table 7.3 (v1eq and dkeq, respectively). The v1eq result is closer to the
116
Table 7.1. E ect of Energy Exchange Models for FIRE II at 1631 sec Run Flux, W/cm2 (0.31-5.4 eV) Identi er Atomic Mol. Atomic Total Cont. Band Line
ight :16 20% v110 .1E-4 .54 .4E-3 .54 v155 .7E-5 2.0 .5E-2 2.0 v173 .1E-4 1.1 .9E-3 1.1 v273 .1E-4 3.2 .2E-1 3.2 v373 .1E-4 .41 .5E-3 .41 dk73 .2E-3 1.7 .5E-1 1.7 v1732evs .1E-4 .72 .5E-3 .73
Table 7.2. E ect of Energy Exchange Models for FIRE II at 1634 sec Run Flux, W/cm2 (0.31-5.4 eV) Identi er Atomic Mol. Atomic Total Cont. Band Line
ight 8:2 20% v110 .2E-1 17.7 2.7 20.3 v155 .4E-1 19.1 3.8 22.9 v173 .3E-1 17.7 3.3 21.1 v273 .7E-1 22.5 5.2 27.8 v373 .1E-1 17.6 2.2 19.9 dk73 .3 3.1 7.9 11.4 v1732evs .1 12.1 6.0 18.3
ight result and is therefore shown on Fig. 7.12. The agreement between this predicted spectrum and the ight spectrum is remarkable. (Recall that the spectral radiometer did not measure below 0.3 m, so there is nothing to compare with in this range.) Thus, examining the radiation spectra at this condition suggests that Case C is in fact in thermal equilibrium, or at least that the distribution of energy between modes is more nearly in equilibrium than was suggested by the other nonequilibrium CFD models.
117
Table 7.3. E ect of Energy Exchange Models for FIRE II at 1637.5 sec Run Flux, W/cm2 (0.31-5.4 eV) Identi er Atomic Mol. Atomic Total Cont. Band Line
ight 81:7 20% v110 3.8 33.5 45.3 82.7 v155 3.9 38.9 46.2 89.0 v173 3.9 35.4 45.7 84.9 v273 4.1 38.8 49.2 92.1 v373 3.8 33.6 44.5 81.9 dkeq 5.2 18.8 99.9 123.9 v1eq 6.5 5.8 75.6 87.8 v1732evs 4.6 20.8 52.0 77.4 v1733dis 4.6 15.5 52.6 72.8
118
8 Conclusions
8.1 Accomplishments A new method, LORAN, was developed to predict gas radiation in conditions of thermochemical nonequilibrium. This method includes a moderately detailed radiative spectrum employing the smeared band model for molecular radiation. It is intended to provide radiation predictions with an accuracy between that of detailed line by line models and what is obtainable from highly approximate but very fast step models. The optimization of the method to provide the maximum detail for the minimum of computational e ort is described. Representative results comparing the method to available ground and ight data indicate that LORAN predicts radiation with accuracy similar to that of Park's NEQAIR code [20] and the nonequilibrium method of Carlson et al. [35]. Two options for radiative transport were incorporated in LORAN. The rst is the traditional tangent slab method, in which the radiating medium is assumed to be one-dimensional. A second transport method was developed without invoking this approximation by applying a modi ed di erential approximation (MDA). Predictions of the radiative heat ux to the wall from the two transport methods were compared and show qualitative agreement with uxes from the MDA method about 20 to 25 percent lower than from the tangent slab. The pro les suggest that the boundary conditions of the MDA method might be improved. The variation of the divergence
119 of the radiative heat ux, appearing in the energy equation to couple the radiation and ow eld properties, was also examined and found to be much smoother in the MDA method. In addition to providing more accurate coupled results, this feature of the MDA method has the potential to enhance the stability of coupled solutions. A sensitivity study was performed using LORAN and ow eld solutions from LAURA to assess the e ect that various models used in nonequilibrium CFD codes have on radiation predictions. The results indicate that radiation is a very sensitive indicator of phenomena occurring in nonequilibrium ow elds. This fact may be used in conjunction with currently available and future ight and ground-based data to improve the modeling of such things as exchange between energy modes in thermochemical nonequilibrium, and chemical reaction rates.
8.2 Future Work Opportunities and needs for additional work have been identi ed in several areas. LORAN makes use of the NEQAIR method and data set for the computation of electronic energy level populations in nonequilibrium. This QSS algorithm incorporates assumptions about the rate of change of energy level populations which are known to be violated in high gradient ow regions. The rate data which it uses are also known to contain uncertainties. Both of these areas of potential error should be investigated in more detail. Many of the radiative properties used in LORAN are uncertain, so the radiative heating values it predicts can only be regarded as approximate. An e ort needs to be
120 undertaken to reduce these uncertainties. To do so may require new measurements to be made beyond those already available in the literature. In addition, corrections to the bound-free and free-free atomic continuum absorption coecients accounting for deviations from a hydrogenic atomic structure could be included, though an initial inquiry suggests that these corrections have little e ect. The multi-dimensional transport algorithm presented here is only a rst iteration. Several aspects of this algorithm will be developed further in the future. These include the application of the method to cases when the absorption coecient corrected for induced emission is nonpositive, the optimum selection of the underrelaxation parameter for each spectral point, and the identi cation of improved boundary conditions. Additional work is needed to make LORAN a practical tool for computing nonequilibrium ow elds with coupled radiation, especially when coupling e ects are significant and many iterations are required to converge the coupled solution. Work is continuing to further reduce the time required to compute radiation by applying vectorization and other ecient strategies.
121
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125 Journal, 9(1):122{130, January 1971.
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[33] W.E. Nicolet. User's Manual for the Generalized Radiation Transfer Code (RAD/EQUIL). NASA CR 116353, October 1969. [34] Kenneth Sutton. Characteristics of Coupled Nongray Radiating Gas Flows with Ablation Products E ects About Blunt Blunt Bodies During Planetary Entries.
PhD thesis, North Carolina State University, Raleigh, North Carolina, 1973. [35] L.A. Carlson and T.A. Gally. NonequilibriumChemical and Radiation Coupling Phenomena in AOTV Flow elds. AIAA Paper 91-0569, January 1991. [36] Chul Park and Frank S. Milos. Computational Equations for Radiating and Ablating Shock Layers. AIAA Paper 90-0356, January 1990. [37] J.N. Moss and J.M. Price. Direct Simulation of AFE Forebody and Wake Flow with Thermal Radiation. NASA TM 100673, September 1988. [38] G.A. Bird. Nonequilibrium Radiation During Re-entry at 10 km/s. AIAA Paper 87-1543, June 1987.
126 [39] Ann B. Carlson. Direct Simulation Monte Carlo With Ionization and Radiation. PhD thesis, North Carolina State University, 1990. [40] Ann B. Carlson and H. A. Hassan. Radiation Modeling with Direct Simulation Monte Carlo. AIAA Paper 91-1409, June 1991. [41] J.D. Teare, S. Georgiev, and R.A. Allen. Radiation from the Non-Equilibrium Shock Front. Research Report 112, AFCRL 937, Avco-Everett Research Laboratory, October 1961. [42] Richard A. Allen, J.C. Keck, and J.C. Camm. Non-Equilibrium Radiation from Shock Heated Nitrogen and a Determination of the Recombination Rate. Research Report 110, Avco-Everett Research Laboratory, June 1961. [43] Richard A. Allen, P.H. Rose, and J.C. Camm. Nonequilibrium and Equilibrium Radiation at Super-Satellite Re-entry Velocities. Research Report 156, BSDTDR-62-349, Avco-Everett Research Laboratory, September 1962. [44] Richard A. Allen. Nonequilibrium Shock Front Rotational, Vibrational and Electronic Temperature Measurements. NASA CR 58673, August 1964. [45] Richard A. Allen, A. Textoris, and J. Wilson. Measurements of the Free-Bound and Free-Free Continua of Nitrogen, Oxygen, and Air. Research Report 195, Avco-Everett Research Laboratory, September 1964. [46] Bennett Kivel. Chemistry, Ionization and Radiation in the Nonequilibrium
127 Front of Normal Shocks in Air. Technical Report AMP 34, Avco-Everett Research Laboratory, September 1959. [47] William A. Page, Thomas N. Canning, Roger A. Craig, and Jack D. Stephenson. Measurements of Thermal Radiation of Air from the Stagnation Region of Blunt Bodies Traveling at velocities up to 31,000 Feet Per Second. NASA TM X-508, 1961. [48] W.A. Page. Shock-Layer Radiation of Blunt Bodies Traveling at Lunar Return Entry Velocities. Technical report, Presented at the IAS 31st Annual Meeting, New York, January 1963. [49] Thomas N. Canning and William A. Page. Measurements of Radiation from the Flow Fields of Bodies Flying at Speeds up to 13.4 Kilometers per Second. In W.C. Nelson, editor, The High Temperature Aspects of Hypersonic Flow, pages 569{581. The Macmillan Co., New York, 1964. [50] Robert M. Nerem. Measurements of Aerodynamic and Radiative Heating at Super-Orbital Velocities. Technical Report The Aerodynamic Laboratory 15981, Ohio State University, January 1964. [51] Robert M. Nerem. Stagnation Point Heat Transfer in High Enthalpy Gas Flows. Part II: Shock Layer Radiative Emission During Hypervelocity Re-entry. FDL TR 64-41 Part II, March 1964.
128 [52] Robert M. Nerem and George H. Stickford. Stagnation Point Heat Transfer in High Enthalpy Gas Flows. Part I: Convective Heat Transfer in Partially Ionized Air. FDL TR 64-41 Part I, March 1964. [53] Robert M. Nerem. Radiating Flows around Re-entry Bodies. Technical Report The Aerodynamic Laboratory 1941-1, Ohio State University, September 1965. [54] R.M. Nerem and G. Stickford. Shock Tube Studies of Equilibrium Radiation. AIAA Journal, 3(6):1011{1018, June 1965.
[55] R.M. Nerem, L.A. Carlson, and R.A. Golobic. Radiation-Gas-Dynamic Coupling Behind Re ected Shock Waves in Air. Technical report, Presented at the 12th International Congress on Applied Mechanics held at Stanford University, California, August 1968. [56] R.M. Nerem, L.A. Carlson, and J.E. Hartsel. Chemical Relaxation Phenomena Behind Normal Shock Waves in a Dissociated Freestream. AIAA Journal, 5(5):910{916, May 1967. [57] Robert A. Golobic and Robert M. Nerem. Shock Tube Measurements of EndWall Radiative Heat Transfer in Air. AIAA Journal, 6(9):1741{1746, September 1968. [58] Robert A. Golobic and Robert M. Nerem. Shock Tube Studies of Radiative Transfer E ects in Air and Xenon. Technical report, NASA-LaRC Symposium
129 on Hypervelocity Radiating Flow elds for Earth and Planetary Entries, Hampton, VA, January 1971. [59] R.M. Nerem, R.A. Golobic, and J.B. Bader. Laboratory Studies of Radiative Transfer E ects in Shock-Heated Gases. In Shock Tube Research: Proceedings of the Eighth International Shock Tube Symposium, pages 19/1{19/19. Chapman
and Hall, London, July 1971. [60] Robert M. Nerem. Radiative Transfer E ects Behind Re ected Shock Waves. Final Report Grant NGL-36-008-106, March 1972. [61] Norman R. Richardson. Project FIRE Instrumentation for Radiative Heating and Related Measurements. NASA TN D-3646, 1966. [62] John H. Lewis, Jr. and William I. Scallion. Flight Parameters and Vehicle Performance for Project FIRE Flight II, Launched May 22, 1965. NASA TN D-3569, August 1966. [63] Hans R. Griem. Spectral Line Broadening by Plasmas. Academic Press, New York, 1974. [64] William A. Page, Dale L. Compton, William J. Borucki, Donald L. Ci one, and David M. Cooper. Radiative Transfer in Inviscid Non-Adiabatic Stagnation Region Shock Layers. AIAA Paper 68-784, June 1968. [65] G. Herzberg. Molecular Spectra and Molecular Structure. I. Spectra of Diatomic
130 Molecules. D. Van Nostrand Co., Inc., Princeton, New Jersey, 1950.
[66] W.L. Wiese, M.W. Smith, and B.M. Glennon. Atomic Transition Probabilities; Volume I: Hydrogen Through Neon. NSRDS-NBS 4. National Bureau of
Standards, May 1966. [67] Chul Park. Nonequilibrium Hypersonic Aerothermodynamics. Wiley, New York, 1990. [68] Ellis Whiting, 1989. Private Communication. [69] Peter A. Gno o, Roop N. Gupta, and Judy L. Shinn. Conservation Equations and Physical Models for Hypersonic Air Flows in Thermal and Chemical Nonequilibrium. NASA TP 2867, February 1989. [70] E. M. Sparrow and R. D. Cess. Radiation Heat Transfer. Brooks/Cole Publishing Company, Belmont, California, 1966. [71] Peter A. Gno o. Upwind-Biased Point-Implicit Relaxation Strategies for Viscous Hypersonic Flows. AIAA Paper 89-1972, June 1989. [72] Chul Park. Assessment of Two-Temperature Kinetic Model for Ionizing Air. Journal of Thermophysics and Heat Transfer, 3(3):233{244, July 1989.
[73] Surendra P. Sharma, Walter D. Gillespie, and Scott A. Meyer. Shock Front Radiation Measurements in Air. AIAA Paper 91-0573, January 1991.
131 [74] John W. Bond, Jr., Kenneth M. Watson, and Jasper A. Welch, Jr. Atomic Theory of Gas Dynamics. Addison-Wesley, Reading, Massachusetts, 1965.
[75] Peter A. Gno o. Asynchronous, Macrotasked Relaxation Strategies For the Solution of Viscous Hypersonic Flows. AIAA Paper 91-1579, June 1991. [76] Dona L. Cauchon. Radiative Heating Results from the FIRE II Flight Experiment at a Reentry Velocity of 11.4 Kilometers per Second. NASA TM X-1402, July 1967. [77] Dona L. Cauchon, Charles W. McKee, and Elden S. Cornette. Spectral Measurements of Gas-Cap Radiation During Project FIRE Flight Experiments at Reentry Velocities Near 11.4 Kilometers per Second. NASA TM X-1389, October 1967. [78] Elden S. Cornette. Forebody Temperatures and Calorimeter Heating Rates Measured During Project FIRE II Reentry at 11.35 Kilometers per Second. NASA TM X-1305, November 1966. [79] Richard C. Dingeldein. Radiative and Total Heating Rates Obtained from Project FIRE Reentries at 37,000 Feet per Second. Technical report, December 1965. Paper presented at the AFFDL/ASSET Advanced Lifting Reentry Technology Symposium. [80] Robert A. Mitcheltree and Peter A. Gno o. Thermochemical Nonequilibrium Issues for Earth Reentry of Mars Mission Vehicles. AIAA Paper 90-1698, June
132 1990. [81] F. McNeil Cheatwood, Fred R. DeJarnette, and H. Harris Hamilton, II. Geometrical Description for a Proposed Aeroassist Flight Experiment Vehicle. NASA TM- 87714, July 1986. [82] J.N. Moss, G.A. Bird, and V.K. Dogra. Nonequilibrium Thermal Radiation for an AFE Vehicle. AIAA Paper 88-0081, January 1988. [83] Peter A. Gno o. Code Calibration Program in Support of the Aeroassist Flight Experiment. Journal of Spacecraft & Rockets, 27(2):131{142, March-April 1990. [84] H. Harris Hamilton, II, Roop N. Gupta, and Jim J. Jones. Flight StagnationPoint Heating Calculations on Aeroassist Flight Experiment Vehicle. Journal of Spacecraft & Rockets, 28(1), January-February 1991.
[85] Robert B. Greendyke, 1991. Private Communication. [86] R.B. Greendyke and L.C. Hartung. An Approximate Method for the Calculation of Nonequilibrium Radiative Heat Transfer. Journal of Spacecraft & Rockets, 28(1), January-February 1991.
[87] Chul Park. Assessment of Two-Temperature Kinetic Model for Ionizing Air. AIAA Paper 87-1574, June 1987. [88] Surendra P. Sharma, Winifred M. Huo, and Chul Park. The Rate Parameters for Coupled Vibration-Dissociation in a Generalized SSH Approximation -
133 Schwartz, Slawsky, and Herzfeld. AIAA Paper 88-2714, June 1988. [89] R.C. Millikan and D.R. White. Systematics of Vibrational Relaxation. Journal of Chemical Physics, 39(12):3209{3213, December 1963.
[90] Chul Park. Problems of Rate Chemistry in the Flight Regimes of Aeroassisted Orbital Transfer Vehicles. In H.F. Nelson, editor, Progress in Astronautics and Aeronautics: Thermal Design of Aeroassisted Orbital Transfer Vehicles,
volume 96, pages 511{537. AIAA Paper, 1985. [91] Surendra P. Sharma and Chul Park. Survey of Simulation and Diagnostic Techniques for Hypersonic Nonequilibrium Flows. Journal of Thermophysics, 4(2):129{142, April 1990. [92] Peter A. Gno o, 1991. Private Communication. [93] Chul Park. Two-Temperature Interpretation of Dissociation Rate Data for N2 and O2. AIAA Paper 88-0458, January 1988. [94] Michael G. Dunn and Sang-Wook Kang. Theoretical and Experimental Studies of Reentry Plasmas. NASA CR 2232, 1973. [95] Chul Park, John T. Howe, Richard L. Ja e, and Graham V. Candler. ChemicalKinetic Problems of Future NASA Missions. AIAA Paper 91-0464, January 1991.
134 [96] Roop N. Gupta, Jerrold M. Yos, Richard A. Thompson, and Kam-Pui Lee. A Review of Reaction Rates and Thermodynamic and Transport Properties for an 11-Species Air Model for Chemical and Thermal Nonequilibrium Calculations to 30 000 K. NASA RP 1232, August 1990. [97] Robert A. Mitcheltree. A Parametric Study of Dissociation and Ionization Models at 12 km/sec. AIAA Paper 91-1368, June 1991. [98] L.A. Carlson and T.A. Gally. The E ect of Electron Temperature and Impact Ionization on Martian Return AOTV Flow elds. AIAA Paper 89-1729, June 1989. [99] Peter A. Gno o. An Upwind-Biased, Point-Implicit Relaxation Algorithm for Viscous, Compressible Perfect-Gas Flows. NASA TP 2953, February 1990. [100] Dale A. Anderson, John C. Tannehill, and Richard H. Pletcher. Computational Fluid Mechanics and Heat Transfer. Hemisphere Publishing Corporation, New
York, 1984.
135
A Gaussian Units Much of the work in radiation has historically been done in the Gaussian system of units. This system introduces the particular oddity of measuring the electron charge in statcoulombs. As this particular unit of measure is uncommon, the following information is provided for the reader. In the Gaussian system of units: c=2.9979x1010
cm/sec
speed of light in vacuum
e=4.80286x10?10 statcoul
electron charge
h=6.6262x10?27
erg-sec
Planck's constant
k=1.3807x10?16
erg/K
Boltzmann's constant
me=9.1095x10?28 g
electron mass.
The following relations hold among several of these quantities: The Bohr radius, a0, is given by 2 a0 = 42hm e2 = 0:52918x10?8 cm
e
(A:1)
The ne structure constant, , a dimensionless quantity, is ?3 = 2e = 7 : 292 x 10 hc 2
(A:2)
136
B Finite Volume Formulation of Radiative Transport B.1 Finite Volume Development
The following development is for the modi ed di erential approximation for radiative transport as reported by Modest [27]. The method has been reexpressed for a non-gray, non-scattering and nonequilibrium medium. The governing equations for the medium ux and incident intensity are:
5~ ~qR = 4je ? 0 G
(B:1)
~ G = ?30 ~qR 5
(B:2)
In what follows the R subscript on the radiative heat ux will be dropped for clarity. The rst of these equations is a scalar equation, while the second is a vector equation. There are accordingly four unknowns for each spectral frequency chosen: G , qx , qy , and qz . The only quantity required for coupling to a Computational Fluid Dynamics
~ ~qR. This quantity (CFD) solution, however, is the divergence of the radiative ux, 5 is given as a function of G by integrating Eq. B.1 over frequency, so that a solution for the single unknown G at each spectral frequency is sucient. To obtain a form of these equations suitable for a nite-volume solution, rst rearrange Eq. B.2:
15 ~ 0 G = ?3~q
(B:3)
137 Now take the dot product of this equation to allow the substitution of Eq. B.1: ! 1 ~ ~q = ?12je + 30 G (B:4) 5~ 0 5~ G = ?35
So now rearranging,
! 1 5~ 0 5~ G = 30 G ? 12je
(B:5)
In the nite volume approach, this is now to be integrated over the volume of a single grid cell, V .
ZZZ
! ZZZ 1 0 G ? 12j e ) dV V (30 G ? 12j e ) ~ 05 ~ G dV = 5 (3 V V (B:6)
Then by application of Gauss' theorem (the divergence theorem), the volume integral on the left-hand-side can be transformed to a surface integral. ! ZZ 1 ~ G n^ da = V (30 G ? 12je ) 5 0 S
(B:7)
The integral is performed by assuming that the absorption coecient 0 and the
~ G are constant on each cell face. The integral gradient of the incident intensity 5 then becomes a summation over the six faces of the cell. The notation used for the cell geometry is that of Gno o [99], where I , J , and K denote cell centers, and i, j , and k denote cell faces. Then ! " 15 ~ 0 G i+1 n^ i+1ai+1 ? 2 ! 1 ~ G n^j+1 aj+1 ? + 4 0 5
j +1
# ! 15 ~ 0 G i n^ iai J;K 3 ! 15 ~ 5 0 G j n^j aj I;K
138 +
"
15 ~ 0 G
# ! 1 ~ n^ k+1 ak+1 ? 0 5G n^k ak k+1 k I;J 0 e = V (3 G ? 12j )I;J;K
!
(B.8)
For the nongray gas found in a shock layer, the radiative properties 0 and je vary over orders of magnitude at various locations and frequencies. To minimize the numerical diculties that this variation can introduce, some kind of normalization is desired. De ne
and
P je j = Ncells cells
(B:9)
P 0 = Ncells cells
(B:10)
Both these quantities are constant for each frequency considered, and provide a measure of the approximate magnitude of the emission and absorption at that frequency. Figure B.1 is an example of the variation of these average coecients for the Mars return test case presented in Ch. 6 and demonstrates the wide variation in radiation properties for various frequencies in a single ow eld. Proceeding with the nite volume development, now divide Eq. B.8 by j on both sides, and also multiply terms containing G by 1 in the form of = . De ne a new variable ? = G =(j ). Then Eq. B.8 becomes: # ! ! " 5 ~ ~ 0 ? i+1 n^ i+1ai+1 ? 0 5? i n^ iai J;K 3 2 ! ! ~ ? ~ ? n^j aj 5 n^j+1 aj+1 ? 0 5 + 4 0 5
j +1
j
I;K
Space-averaged κ ν’ and jνe
139 10 5 10 4 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 0.0
κν jν
2.5
5.0
7.5
10.0
12.5
15.0
17.5
hν, eV
Figure B.1. Variation of and j for a Mars Return Flow eld "
~ ? + 0 5
# ! ~ n^ k+1 ak+1 ? 0 5? n^k ak k+1 k I;J e! j = V 3 0 ? ? 12 j I;J;K
!
(B.11)
This formulation reduces the variation of the unknown, ? , and of the coecients of the equation, making the numerical solution easier. Now consider the j and j + 1 cell faces in an axisymmetric ow. The gradient of ? has no component in the circumferential direction, while the surface normal n^ on these faces is entirely in the circumferential direction. Therefore the dot product of these two quantities is zero and the terms on these faces drop out. This is a mathematical expression of the fact that there is no ux through these cell faces in an axisymmetric ow. The j and J subscripts then become super uous, and will be omitted in the remainder of this development.
140 Since the ow eld grid is not orthogonal, it is necessary to use a generalized transformation to obtain nite di erence expressions for the gradients in Eq. B.11. The elements of such a transformation are repeated below from Anderson et al. [100, Sec. 5-6.2], specialized to the case of an axisymmetric ow eld. Second order accurate central di erence expressions are used to obtain the necessary partial derivatives, where
, , and are the coordinates of the orthogonal and uniformly spaced computational grid corresponding to the i, j , and k grid directions, respectively.
x = xi+1;k ?2 xi?1;k
(B:12)
x = 0
(B:13)
x = xi;k+1 ?2 xi;k?1
(B:14)
y = 0
(B:15)
y = yi;j+1;k ?2 yi;j?1;k = xi;k sin
(B:16)
= 2: arctan xy jcell corner
(B:17)
y = 0
(B:18)
z = zi+1;k ?2 zi?1;k
(B:19)
z = 0
(B:20)
z = zi;k+1 ?2 zi;k?1
(B:21)
where
141 At the boundaries rst order backward di erences are applied, except for the axis of symmetry boundary. At this boundary the coordinates of a \ghost" cell can be obtained through re ection. Second order accurate central di erencing can therefore be applied at this boundary. The Jacobian of the transformation can then be obtained from:
J^ = x y z ?1 x y z
(B:22)
while the metrics of the transformation are given by: ^ z x = Jy
(B:23)
y = 0
(B:24)
^ y z = ?Jx
(B:25)
x = 0
(B:26)
y = J^(x z ? x z )
(B:27)
z = 0
(B:28)
^ z x = ?Jy
(B:29)
y = 0
(B:30)
^ y z = Jx
(B:31)
The rst derivatives which appear in the gradient terms are then obtained from
@ = @ + @ @x x @ x @
(B:32)
142
@ = @ @y y @ @ = @ + @ @z z @ z @
(B:33) (B:34)
so the gradient can be expanded as: ! ! ~5? = x @ ? + x @ ? ^i + y @ ? ^j + z @ ? + z @ ? k^ @ @ @ @ @
(B:35)
The y-derivative can be omitted since there is no circumferential variation of ? in the case of axisymmetric ow. Collecting terms then yields
~ ? = x^i + z k^ @ ? + x^i + z k^ @ ? = 5 ~ @ ? + 5 ~ @ ? 5 @ @ @ @
(B:36)
Now note that the combination n^ a is just the directed area of the cell face, or ~a. The nite volume expression, Eq. B.11, can now be reformulated as: ! 5 @ ? @ ? ~ ~ 0i+1;K i+1;K @ i+1;K + 5i+1;K @ i+1;K ~ai+1;K ! @ ? @ ? ~ ~ ~ai;K ? 0 5i;K @ + 5i;K @ i;K i;K i;K ! @ ? @ ? ~ ~ + 0 ~aI;k+1 + 5I;k+1 @ 5I;k+1 @ I;k+1 I;k+1 I;k+1 ! @ ? @ ? ~ ~ ? 0 5I;k @ + 5I;k @ ~aI;k I;k I;k I;k e! j 0 = VI;K 3 ? ? 12 j I;K
(B.37)
The radiative properties 0 and je are known at cell centers (I; K ). The required values of 0 at the cell faces can be obtained by averaging the values at the two nearest cell centers. Then for instance 1
0i+1;K
0 1 1 1 1 = 2 @ 0 + 0 A I +1;K I;K
(B:38)
143 This average has been found to produce better stability than the alternate form with 2=(0I+1;K + 0I;K ) because of the extreme variation of the absorption coecient near the freestream boundary in particular. The rst derivatives of ? in computational space at the cell faces are obtained by second-order accurate central di erences, as follows:
@ ? @ i+1;K = (?I+1;K ? ?I;K )
(B:39)
@ ? 1 ?I+1;K+1 ? ?I+1;K?1 + ?I;K+1 ? ?I;K?1 = @ i+1;K 2 2 2
!
(B:40)
The next two di erences are the same but with a shift in the i and I indices:
@ ? = (? ? ? I;K I ?1;K ) @ i;K @ ? = 1 ?I;K+1 ? ?I;K?1 + ?I?1;K+1 ? ?I?1;K?1 @ i;K 2 2 2
(B:41)
!
(B:42)
Along the normal cell faces:
@ ? 1 ?I+1;K+1 ? ?I?1;K+1 + ?I+1;K ? ?I?1;K = @ I;k+1 2 2 2
!
@ ? @ I;k+1 = (?I;K+1 ? ?I;K )
(B:43) (B:44)
And again the next two di erences are the same except for a shift in the k and K indices:
@ ? = 1 ?I+1;K ? ?I?1;K + ?I+1;K?1 ? ?I?1;K?1 @ I;k 2 2 2 @ ? = (? ? ? I;K I;K ?1 ) @ I;k
!
(B:45) (B:46)
144 All these expressions can now be substituted back into Eq. B.37, to obtain:
h5 ~ 0i+1;K i+1;K (?I+1;K ? ?I;K ) !# ? ? ? ? ? ? 1 I +1 ;K +1 I +1 ;K ? 1 I;K +1 I;K ? 1 ~ i+1;K ~ai+1;K + +5 2 2 2 h ? 0 5~ i;K (?I;K ? ?I?1;K ) i;K !# ? ? ? ? ? ? 1 I;K +1 I;K ? 1 I ? 1 ;K +1 I ? 1 ;K ? 1 ~ i;K ~ai;K +5 + 2 2 2 " ! ?I+1;K+1 ? ?I?1;K+1 ?I+1;K ? ?I?1;K 1 ~ + 0 5I;k+1 2 + 2 2 I;k+1 ~ I;k+1(?I;K+1 ? ?I;K )i ~aI;k+1 +5 ! " ?I+1;K ? ?I?1;K ?I+1;K?1 ? ?I?1;K?1 1 ~ + ? 0 5I;k 2 2 2 I;k e! i j ~ I;k (?I;K ? ?I;K?1 ) ~aI;k = VI;K 3 0 ? ? 12 +5 j I;K
(B.47)
Carrying out the dot products and collecting terms results in:
d1?I+1;K+1 + d2?I+1;K + d3?I+1;K?1 + d4?I;K+1 + d5?I;K + d6?I;K?1 jeI;K +d7?I?1;K+1 + d8?I?1;K + d9?I?1;K?1 = ?12VI;K j
(B.48)
where
~ i+1;K ~ai+1;K + 1 0 5 ~ I;k+1 ~aI;k+1 d1 = 14 0 5 4 i+1;K
I;k+1
(B.49)
~ i+1;K ~ai+1;K + 1 0 5 ~ I;k+1 ~aI;k+1 ? 1 0 5 ~ I;k ~aI;k (B.50) d2 = 0 5 4 4 i+1;K
I;k+1
I;k
~ i+1;K ~ai+1;K ? 1 0 5 ~ I;k ~aI;k d3 = ? 14 0 5 4 i+1;K
I;k
(B.51)
145
~ i+1;K ~ai+1;K ? 1 0 5 ~ i;K ~ai;K + 0 5 ~ I;k+1 ~aI;k+1 (B.52) d4 = 14 0 5 4 i+1;K
i;K
I;k+1
~ i+1;K ~ai+1;K ? 0 5 ~ d5 = ? 0 5 i;K i;K ~ai;K i+1;K ~ I;k+1 ~aI;k+1 ? 0 5 ~ I;k ~aI;k ? 3VI;K 0I;K ? 0 5 I;k I;k+1
(B.53)
~ i+1;K ~ai+1;K + 1 0 5 ~ i;K ~ai;K + 0 5 ~ I;k ~aI;k (B.54) d6 = ? 14 0 5 4 i+1;K
i;K
I;k
~ i;K ~ai;K ? 1 0 5 ~ I;k+1 ~aI;k+1 d7 = ? 41 0 5 4
(B.55)
~ i;K ~ai;K ? 1 0 5 ~ I;k+1 ~aI;k+1 + 1 0 5 ~ I;k ~aI;k d8 = 0 5 4 4
(B.56)
~ i;K ~ai;K + 1 0 5 ~ I;k ~aI;k d9 = 14 0 5 4
(B.57)
i;K
i;K
I;k+1
I;k+1
i;K
I;k
I;k
The dot products of the metrics with the cell face directed areas which appear in these coecients are determined by the grid geometry. These can be written as follows:[99]
# V ( ~ a + ~ a ) + V ( ~ a + ~ a ) I;K i ? 1 ;K i;K I ? 1 ;K i;K i +1 ;K ~ i;K ~ai;K = ~ai;K = 5 4VI;K VI ?1;K VI;K (axi?1;K axi;K + azi?1;K azi;K + axi;K axi;K + azi;K azi;K ) 4VI;K VI ?1;K V (a a + a a + a a +a a ) + I ?1;K xi;K xi;K zi;K4V zi;KV xi+1;K xi;K zi+1;K zi;K "
I;K I ?1;K
(B.58)
# " ~5I;k ~aI;k = VI;K (~ai;K?1 + ~ai+1;K?1) + VI;K?1 (~ai;K + ~ai+1;K ) ~aI;k = 4VI;K VI;K?1 VI;K (axi;K?1 axI;k + azi;K?1 azI;k + axi+1;K?1 axI;k + azi+1;K?1 azI;k ) 4VI;K VI;K?1 V (a a + a a + a a +a a ) + I;K?1 xi;K xI;k zi;K4V zI;kV xi+1;K xI;k zi+1;K zI;k (B.59) I;K I;K ?1
146
# " ~5i;K ~ai;K = VI;K (~aI ?1;k + ~aI ?1;k+1) + VI ?1;K (~aI;k + ~aI;k+1) ~ai;K = 4VI;K VI ?1;K VI;K (axI?1;k axi;K + azI?1;k azi;K + axI?1;k+1 axi;K + azI?1;k+1 azi;K ) 4VI;K VI ?1;K V (a a + a a + a a + a a ) + I ?1;K xI;k xi;K zI;k4V zi;KV xI;k+1 xi;K zI;k+1 zi;K (B.60) "
I;K I ?1;K
# V ( ~ a + ~ a ) + V ( ~ a + ~ a ) I;K I;k ? 1 I;k I;K ? 1 I;k I;k +1 ~ I;k ~aI;k = 5 ~aI;k = 4VI;K VI;K?1 VI;K (axI;k?1 axI;k + azI;k?1 azI;k + axI;k axI;k + azI;k azI;k ) 4VI;K VI;K?1 V (a a + a a + a a + a a ) + I;K?1 xI;k xI;k zI;k4V zI;kV xI;k+1 xI;k zI;k+1 zI;k I;K I;K ?1
(B.61)
Equation B.48 is a matrix equation for the unknowns ?I;K of the general form
Ax = b. The matrix A and the vector b are functions only of the geometry and the absorption and emission coecients. The numerical solution of this type of equation can be obtained by a number of methods, including direct inversion of the matrix, and various relaxation methods such as point Jacobi, point Gauss-Seidel, line Jacobi or line Gauss-Seidel algorithms. The line Gauss-Seidel algorithm has been selected and applied along each normal grid line to capture the dominant gradients in the radiative properties. In this approach, all the ? terms along the I grid line are at the n + 1 time level, while all I +1 terms are at the n time level. Terms with I ? 1 are known at time level n +1 from the solution of the previous line. The d coecients, as mentioned above, are functions only of the geometry and the absorption coecient. Since both these quantities are constant for a given transport solution, the d coecients need only be calculated once. The line Gauss-Seidel algorithm results in a tridiagonal matrix equation to be
147 solved for each normal grid line. The solution of such an equation can easily be obtained using the Thomas algorithm [100] with speci cation of appropriate boundary conditions as discussed below. To ensure convergence for this problem, which is a nonlinear, elliptic equation, underrelaxation is recommended. In fact it is found to be necessary in this application. It is introduced by de ning a corrected value for the update of ? as +10 = ?n + r(?n+1 ? ?n ) = (1 ? r)?n + r?n+1 ?nI;K I;K I;K I;K I;K I;K
(B:62)
where r is the relaxation parameter and is less than one for underrelaxation. The +1 is obtained by solving Eq. B.48. Incorporating this expression in expression for ?nI;K
Eq. B.62 and rearranging to the line Gauss-Seidel form results in
n +1 + ?n+10 + d r ?n+1 = (1 ? r)?n ? r d ?n d6 dr ?nI;K 1 I +1;K +1 + d2 ?I +1;K 4 I;K I;K +1 I;K d ? 1 d 5 5 5 jeI;K ! n n +1 n +1 n +1 +d3?I+1;K?1 + d7?I?1;K+1 + d8?I?1;K + d9?I?1;K?1 + 12VI;K j (B.63) The selection of r is discussed in Sec. B.3 below.
B.2 Boundary Conditions Figure B.2 shows the essential features of the radiation grid. Properties at any interior point (denoted by lled circles) can be obtained by the solution of Eq. B.63 above. The un lled circles at the left represent \ghost" points for the symmetry boundary condition at the axis of symmetry. Symmetry boundary conditions also exist in and out of the page on the \pie slice" sides (see Fig. B.3), but these are
148 implicit in the axisymmetric formulation. The triangles along the upper edge of the grid represent the outermost grid cell which forms the freestream boundary. The downward-pointing triangles along the lower edge at right are \ghost" points for the out ow boundary. Finally, the barely visible un lled squares along the lower edge of the grid represent the center of the surface elements. Each of these boundaries is discussed separately below. 0.2 5 -0.00 -0.2 5
z, m
-0.50 -0.75 -1.00
-1.2 5 -0.0
0.5
1.0
1.5
2 .0
2 .5
3.0
x, m
Figure B.2. Essential Features of Axisymmetric Radiation Grid
B.2.1 Axis of Symmetry Boundary This is a symmetry boundary, at which re ection boundary conditions should apply. Therefore for the un lled circles in Fig. B.2 along this boundary: ? (ghost; K ) = ? (1; K )
(B:64)
149 1.5 1.0 0.5
y, m
0.0
-0.5 -1.0 -1.5 0.0
0.5
1.0
1.5
2 .0
x, m
Figure B.3. Top View of Axisymmetric Radiation Grid This boundary is a singularity in the computational grid, however, where one surface of a grid cell has zero area. Referring back to Eq. B.37, this means the term on the second line disappears. The two remaining -derivative terms at the I grid line are replaced by rst order backward di erences to avoid di erencing across the singularity[92]. Then
@ ? 1 ? ? ? + ? ? ? = @ I;k+1 2 I+1;K+1 I;K+1 I+1;K I;K @ ? = 1 ? ? ? + ? ? ? @ I;k 2 I+1;K I;K I+1;K?1 I;K?1 Substituting, Eq. B.47 becomes
h5 ~ 0i+1;K i+1;K (?I+1;K ? ?I;K ) !# ? ? ? ? ? ? 1 I;K +1 I;K ? 1 I +1 ;K +1 I +1 ;K ? 1 ~ i+1;K + ~ai+1;K +5 2 2 2
(B:65) (B:66)
150
5~ I;k+1 21 ?I+1;K+1 ? ?I;K+1 + ?I+1;K ? ?I;K + 0 I;k+1 ~ I;k+1(?I;K+1 ? ?I;K )i ~aI;k+1 +5 ~ ? 0 5I;k 12 ?I+1;K ? ?I;K + ?I+1;K?1 ? ?I;K?1 I;k e! i j 0 ~ I;k (?I;K ? ?I;K?1 ) ~aI;k = VI;K 3 ? ? 12 +5 j I;K
(B.67)
Carrying out the dot products and collecting terms as before allows this equation to be put in the form of Eq. B.48, with
~ i+1;K ~ai+1;K + 1 0 5 ~ I;k+1 ~aI;k+1 d1 = 41 0 5 2 i+1;K
I;k+1
(B.68)
~ i+1;K ~ai+1;K + 1 0 5 ~ I;k+1 ~aI;k+1 ? 1 0 5 ~ I;k ~aI;k (B.69) d2 = 0 5 2 2 i+1;K
I;k+1
I;k
~ i+1;K ~ai+1;K ? 1 0 5 ~ I;k ~aI;k d3 = ? 14 0 5 2 i+1;K
I;k
~ i+1;K ~ai+1;K ? 1 0 5 ~ I;k+1 ~aI;k+1 d4 = 14 0 5 2 i+1;K I;k+1 ~ I;k+1 ~aI;k+1 + 0 5 I;k+1
(B.70)
(B.71)
~ i+1;K ~ai+1;K ? 1 0 5 ~ d5 = ? 0 5 2 I;k+1 I;k+1 ~aI;k+1 i+1;K ~ I;k+1 ~aI;k+1 + 1 0 5 ~ I;k ~aI;k ? 0 5 ~ I;k ~aI;k ? 0 5 2 I;k+1 I;k I;k ?3VI;K 0I;K
(B.72)
~ i+1;K ~ai+1;K + 1 0 5 ~ I;k ~aI;k + 0 5 ~ I;k ~aI;k d6 = ? 14 0 5 2
(B.73)
i+1;K
I;k
I;k
151
d7 = 0
(B:74)
d8 = 0
(B:75)
d9 = 0
(B:76)
With the appropriate values of di along this boundary, then, it can be solved with the same method as the interior points.
B.2.2 Freestream Boundary The freestream \boundary" is transparent to radiation, and only outgoing radiation is considered. The freestream is assumed to be non-emitting. With these assumptions, this boundary can be approximated as a cold wall with complete absorption (" = 1:0). The modi ed di erential approximation is essentially a P-1 method, for which cold wall boundary conditions can be obtained analytically. Modest suggests the use of Marshak's boundary condition, which is ? 2 "2 ? 1 ~q n^ + G = 0
(B:77)
where " is the spectral surface emissivity, and n^ is the outward unit normal vector describing this boundary surface. Replacing ~q in this equation using Eq. B.2 leads to an equation containing only the one variable G : 2 5 ~ 2 ? 1 G0 n^ + G = 0 " 3 Then dividing by (j ) to obtain an equation for ? nally gives 2 5 ~ 2 " ? 1 3?0 n^ + ? = 0
(B:78)
(B:79)
152 Expanding the gradient of ? gives: # " @ ? 2 2 ? 1 5 @ ? ~ ~ n^ @ + 5 n^ @ + ? = 0 30 "
(B:80)
This can be di erenced by expressing the derivatives of ? using a central di erence for the -direction and a rst-order backward di erence for the -direction. Denoting the freestream boundary grid cell by the subscript F , the result is " # ? ? ? 2 5 I +1 ;F I ? 1 ;F ~ I;F n^ I;F ~ I;F n^ I;F ?I;F ? ?I;F ?1 +5 30 2 I;F
+?I;F = 0
(B.81)
where the emissivity for the freestream boundary was set to 1.0 as mentioned above. Collecting terms and arranging them in the line Gauss-Seidel form, with the addition of underrelaxation, gives
~ n+10 +1 + 25 0 ?2r5~ I;F n^ I;F ?nI;F n ^ + 3 I;F I;F ?1 I;F ?I;F =
~ ~ I;F n^I;F (?nI+1;F ? ?nI+1?1;F ) 25I;F n^ I;F + 30I;F (1 ? r)?nI;F ? r5
(B.82)
The dot product terms calculated in Eqs. B.59 and B.61 are at cell faces. To obtain the values at the cell center required in the above expression, an average at the two opposing faces of the cell is taken. Then
~
~
~
~
5~ I;F n^ I;F = 5I;f +1 n^ I;f +1 + 5I;f n^I;f + 54I +1;f +1 n^ I +1;f +1 + 5I +1;f n^ I +1;f 5~ I;F n^ I;F
(B:83) ~ ~ ~ ~ = 5I;f +1 n^ I;f +1 + 5I;f n^I;f + 54I +1;f +1 n^ I +1;f +1 + 5I +1;f n^ I +1;f (B:84)
153 where the subscript f + 1 denotes the freestream cell face. For the freestream boundary at the axis of symmetry, the re ection boundary condition Eq. B.64 for ?I?1;F is used. Using a rst order backward di erence at this point was found to lead to nonphysical results.
B.2.3 Out ow Boundary Along the out ow \boundary", ?I+1;K in Eq. B.47 represents the \ghost" point and must be approximated. This is an arbitrary computational boundary through which radiating gas and radiation (inwardly and outwardly directed) both pass. The
ow eld boundary condition used in LAURA is a zeroth order extrapolation (i.e., the derivatives of the ow variables are assumed to be zero across this boundary). This boundary condition is known to provide better stability than other possible formulations, and so can also be adopted for the radiation method. Therefore, ? (I + 1; K ) ? (I; K )
(B:85)
One alternative possibility is to assume a constant slope at this boundary. To perform an extrapolation of the positive-de nite variable ? , a logarithmic extrapolation is indicated1 ) ? (I + 1; K ) ? ?(I (?I; 1K; K ) 2
(B:86)
This boundary condition was found to work well in some cases and to cause ? (I; K ) to blow up in the shoulder region in others. This undesirable behavior results from 1
Suggested by Gno o
154 inaccuracies and lack of grid resolution near the shoulder of the body in the test cases. Logic was implemented to apply the logarithmic boundary condition except in cases where it would result in an increasing value of ? (I; K ) around the shoulder. Those instances are treated with the zeroth order boundary condition to prevent the solution from blowing up. Even when the solution does blow up at the shoulder, much of the rest of the ow eld remains quite una ected. This suggests that the solution method is robust.
B.2.4 Wall Boundary For the medium only intensity, this boundary appears as a cold wall. The Marshak boundary condition (Eq. B.79) can be di erenced at this boundary by recognizing that the grid lines are normal to the body surface (Fig. B.2 is not to scale). The term
5~ ? n^ is then simply the gradient of ? along the grid lines. This di erence is best expressed in physical coordinates with a rst order forward scheme as ! ?I;1 ? ?I;w 30I;1 "I ? 2 2 ? " ?I;w = 0 r
(B:87)
I
where the subscript w denotes a \ghost" point on the surface of the body, and
q r = (x(I; 1) ? x(I; w))2 + (z(I; 1) ? z(I; w))2
(B:88)
In the Gauss-Seidel format, with the incorporation of underrelaxation, this becomes !# " " 30I;1 "I !# n 1 1 1 + 30I;1 "I 0 n +1 n +1 ?I;w ? r r ?I;1 = (1 ? r) r + 2 2 ? " ?I;w r 2 2?" I
I
(B:89)
155
B.2.5 Radiative Heat Flux to Wall The radiative ux directed to the wall due to emission in the medium can then be found from solving Eq. B.2.
~ G ~q = 3?10 5
(B:90)
In this equation, G is obtained from the solution of the nite volume problem, ? , multiplied by the normalizing factor (j ). To obtain the ux directed at the wall, the dot product of ~q with the wall-directed surface normal is required. Then
~ G n^ w qw = ?~q n^ w = 31 0 5
(B:91)
Again recalling that the grid lines are normal to the body surface, the gradient of G can be expressed with a one-sided di erence along the normal grid lines. As above, this is best done in physical coordinates.
G ?G qw = 31 0 I;2 r I;1
(B:92)
Alternately, a second order forward di erence could be used to represent the gradient. The total heat ux to the wall at each grid location is then obtained by a numerical integration over the spectral result above. This closes the numerical problem.
B.3 Numerics and Convergence
B.3.1 Convergence Criterion
The numerical solution of this set of di erence equations requires the selection of an appropriate criterion to test for convergence. Because the radiative properties vary
156 over many orders of magnitude, and because the grid introduces numerical errors, the usual de nition of the L2 norm for this purpose is not satisfactory. Instead, a local error function is de ned: +1 ? ?n j j?nI;K I;K LI;K = mean(? n)
(B:93)
where the mean of ?n is obtained as the average of the minimum and maximum values of ?nI;K for all I and K on the radiation grid. This de nition reduces the value of
LI;K in regions where ?nI;K is small, relative to the error that would be obtained from the L2 norm. The result is that convergence is reached faster. Though convergence may not be entirely complete at those locations where ?nI;K is small, these locations contribute little to the coupling with the ow eld. Therefore, the lack of complete convergence at these locations is deemed acceptable to reduce the computation time. The actual convergence is determined by the average value of LI;K over the radiation grid. When this average is less than some speci ed value, convergence is assumed to have been reached for that particular frequency . Each frequency is converged individually, since the rate of convergence depends on the magnitude of the optical depth and varies considerably with frequency. A typical convergence history for a single frequency is shown in Fig. B.4. Convergence may be faster or slower for the individual frequencies, depending at least partly on the magnitude of the optical depth at that frequency.
157 10
0
10 -1
L2 Norm
10
-2
10 -3 10
-4
10 -5 10
-6
0
50
100
150
2 00
2 50
300
350
Iteration Number
Figure B.4. Typical Convergence History for MDA Solution
B.3.2 Selection of Relaxation Parameter The relaxation parameter required to obtain a stable solution varies with the magnitude of the optical depth of the ow eld. Very fast convergence can be obtained for optically thick spectral regions using r=1, meaning no underrelaxation. For optically thin regions on the other hand, the solution with r=1 is unstable. The value r=0.5 has been selected as a good compromise. In future, an algorithm might be developed to vary the underrelaxation parameter r as a function of the radiation properties in order to further accelerate the convergence of the complete transport solution.
